I'm going to be pedantic and point out that your question misrepresents Putnam. He wasn't trying to argue for skepticism but to argue against realism. From here:

1) If metaphysical realism is true, then global skepticism is possible
2) If global skepticism is possible, then we can be brains in a vat
3) But we cannot be brains in a vat
4) Thus, metaphysical realism is false

His defence of 3) assumes the causal constraint on reference that was supported by metaphysical realists at the time ("A term refers to an object only if there is an appropriate causal connection between that term and the object"):

1) Assume we are brains in a vat
2) If we are brains in a vat, then "brain" does not refer to brain, and "vat" does not refer to vat (via CC)
3) If “brain in a vat” does not refer to brains in a vat, then “we are brains in a vat” is false
4) Thus, if we are brains in a vat, then the sentence “We are brains in a vat” is false

Regarding 2), the point is that the words "brain" and "vat" as spoken by a BIV person cannot refer to real brains and real vats because there is no appropriate causal connection between real brains/real vats and the BIV's language; instead the words "brain" and "vat" as spoken by a BIV person refer to simulated brains and simulated vats.

I've seen a whole bunch of attempts to prove that we are not brains in vats. Putnam attempts to prove that we can never say that we are brains in vats and have that be true. This is because there isn't a causal connection between the vat in the 'real' world and us in the virtual envatted world. Without this causal connection we cannot refer to vats, to the sentence 'I am a brain in a vat can' not be true.

I don't believe there is a way out of the skeptical hypothesis. Once we take the view that we are somehow experiencing the mental, there is no way for us to use thought (further mental) to think our way to the 'real' world.

I don't believe there is a way out of the skeptical hypothesis. — shmik

By denying the realist's claim that "truth is not reducible to epistemic notions but concerns the nature of a mind-independent reality"¹, which was Putnam's goal.

¹ http://www.iep.utm.edu/brainvat/#H1

. I don't get what that denial achieves
It doesn't prove that someone isn't a brain in a vat.

If you deny the distinction between truth and epistemology then you avoid global skepticism, and if you avoid global skepticism then you avoid the brain-in-a-vat possibility.

For example, if the real world just is the world that we see and if we don't see that we are brains in a vat then we are not brains in a vat.

By that sentence do you mean that we accept semantic externalism?
I don't think the argument can rule out a skeptical hypothesis, just rule out specifying the exact conditions of some skeptical hypothesis.
For instance, I could still say 'it's possible that when I was 12 a mad scientist placed my brain into a vat and I have been living in a vat ever since'. This formulation would allow me to refer to brains in vats.

Also a curiosity of the argument is that if we as a forum got together and decided to envat someone who had just been born. They would be a brain in a vat, but they would never be able to entertain the notion that they were a brain in a vat.

For instance, I could still say 'it's possible that when I was 12 a mad scientist placed my brain into a vat and I have been living in a vat ever since'. This formulation would allow me to refer to brains in vats. — shmik

But given metaphysical realism it's also possible that you were always a brain in a vat. However, given the causal constraint on reference, you could never refer to the real brain in a vat that you are (and always have been), and so realism entails a necessary falsehood, refuting itself.

How can I know that I am really here typing this message and not just a brain in a vat as Putnam challenges us to prove? — Jamesk

You can't know the answer to questions like this with certainty. You can't prove empirical claims.

The issue instead, then, is this: Why should you believe one thing versus another? If one of the candidates for belief is the brain in the vat scenario, what would you take to be a good reason to believe that you're a brain in a vat?

By denying the realist's claim that "truth is not reducible to epistemic notions but concerns the nature of a mind-independent reality"1, which was Putnam's goal. — Michael

Wouldn't you first have to show that there is a mind independent reality?

Who, the realist? Sure. But given that Putnam was trying to provide an argument against realism it's not enough for him to simply demand that the realist prove his position. You can't just assume falsity until proven otherwise as that would be as question-begging as the opposite.

instead the words "brain" and "vat" as spoken by a BIV person refer to simulated brains and simulated vats. — Michael

In order that there could be simulated brains and simulated vats; there must be real brains and real vats. That's the objection I have always had to this silly 'thought experiment'. I'm not at all familiar with Putnam, but something like that objection seems to be what he is presenting

In order that there could be simulated brains and simulated vats; there must be real brains and real vats. — John

Not that it matters to Putnam's argument, but this isn't true. We have simulated unicorns and dragons without there being real unicorns and real dragons.

Once we take the view that we are somehow experiencing the mental, there is no way for us to use thought (further mental) to think our way to the 'real' world. — shmik

But the real world is conceptually articulated through and through; so why would we need to "make our way there"? On the contrary, it cannot be escaped...

That makes no sense. To simulate is to feign reality. If you simulate love you pretend to love when you do not love. If you put a cardboard cutout sheep in the paddock, so that it looks like a real sheep that is a simulated sheep.
What is a simulated dragon?

Put on a virtual reality fantasy game and see.

If there were a lion in such a game we could say that it is simulating a real lion.

If there were a dragon in such a game what exactly is it that you think it could be said to be simulating?

I'm going to be pedantic and point out that your question misrepresents Putnam. — Michael

Re Putnam, by the way, I don't at all agree with his argument, because I don't at all agree with his views of how meaning works.

A dragon.

It was the view that the majority of realists at the time supported, so on that account, assuming the validity of the argument, he successfully refuted their position. Of course, they could respond by denying the causal constraint on reference, but then they'd have to find some alternative way to explain how our words are able to refer to mind-independent things.

However, given the causal constraint on reference, you could never refer to the real brain in a vat that you are (and always have been), and so realism entails a necessary falsehood, refuting itself. — Michael

I wander about this - it's been a while since I've read Putnam so I might try to read an SEP article on it later. Why would realism entail that I always have access to / be able to refer to / mind independent reality?
None of this shows why realism is false, just that a certain way realist talk is false.

But the real world is conceptually articulated through and through; so why would we need to "make our way there"? On the contrary, it cannot be escaped... — John

By this do you just mean some kind of post-Kantian position in which the world is inseparable from us? Holding that cannot refute skepticism, you could still be a BIV.

EDIT: It also entails its own kind of skepticism. If we limit truth so that it's truth for us, we lose the great outdoors of the world apart from us.

Show me a dragon that is real then. If there are no real dragons then it makes no sense to speak of simulated dragons; it's simply nonsensical.

I think it is indutiably true that everything we know and experience is a 'brain state', to use that inelegant expression. (On the one hand, neural reductionism seems to always want to say that; on the other, they don't seem to realise that, if it is true, then 'neural reductionism' is also a brain state! In other words, if you pursue the deflationary tactic of claiming that 'everything is simply patterns of neurochemical reactions', that also applies to that statement, so I can't see how it amounts to a truth claim.)

But I think it can't be denied that the mind is implicated in the knowledge of anything, even of so-called 'mind-independent realities'. But that doesn't mean that the world is 'all in the mind', either; simply that the mental is an inextricable pole or aspect of all experience.

I'm not a great lover of Kant's philosophy, although I do acknowledge its tedious brilliance. And I don't believe I can be a BIV, for the reasons I already presented.

I say we don't have "a world apart from us", and even if there were one, it could only be as nothing to us. The "great outdoors" is the very world we know so well. We lose that just when we incoherently try to reduce it in our imaginations to something utterly alien, unintelligible and meaningless to us, in the interests of an overweeningly dry and empty rationality.

Yes, it's an interesting irony that in order to support the view that the world is nothing more than a mental representation or brain state; a view purportedly antithetical to naive realism; the naively realistic belief that there are real eyes, real nervous systems and real brains is always the unacknowledged but indispensable supporting premise.

Obviously naive realism is true.

Putnam decisively refutes the skeptic idea that we might just be brains in a vat. For example, in Reason, truth, and history (1981).

Apparently it's available online at https://www.archive.org/stream/HilaryPutnam/PutnamHilary-ReasonTruthAndHistory_djvu.txt

We don't need for there to be real dragons for there to be simulated dragons, just as we don't need for there to be a real apocalyptic meteor strike on Earth for there to be a simulated apocalyptic meteor strike on Earth – and, you know, we have such things.

Obviously naive realism is true.

Putnam decisively refutes the skeptic idea that we might just be brains in a vat. For example, in Reason, truth, and history (1981).

Apparently it's available online at https://www.archive.org/stream/HilaryPutnam/PutnamHilary-ReasonTruthAndHistory_djvu.txt — jkop

The irony here is that in Reason, Truth, and History Putnam's intention is to refute realism.

Like a true philosopher Putnam was guided by argument, not intention, and ended up defending naive realism.

No he didn't. His argument concluded that realism is false.