Do we know exactly where the Big Bang took place? Well, yes, you could say that we know exactly where ti took place and still is taking place.
As I see it, which isn't very far, I'd go further than Pedant and say that from our point of view (i.e. as members of what we call the (or this) universe) the Big Bang is everywhere, which is to say: The Big Bang IS the Universe. How, and into what, can it expand then, you ask? It expands by growing more universe. It is rather like a floating soap bubble, only in more than three dimensions: imagine the universe as the analogue of the surface of the bubble, in particular in more dimensions; that is to say that to inhabitants of the surface, there IS NO OTHER UNIVERSE, no inside or outside of the bubble, just surface or skin if you like. Now imagine the bubble warming up (personally I would lay the blame on Carbon Dioxide) and expanding. A pedantic inhabitant might ask: "Where is it expanding into? What is beyond its border?" We say that it doesn't have a border; when it expands it creates more surface without more border, which it can do partly because it really does not have any border to start with.
Now, the Big Bang was the start of the expansion. Remember that whether the bubble was about ten to the minus forty-odd metre in radius, or ten to the plus forty, the whole of its surface was the whole of the universe, so the whole of the universe was the BB, so there was nowhere that wasn't Big Bang.
So, we must take greater care. The Big Bang might be everywhere, including inside each of us (Darned beans!) meaning that we cannot say where it happened, but if we pop that bubble, that is where the bubble pops, even if the pop is a tiny prick. For the first time since time began in Our Universe, we would have created a border to the universe, and a border that would grow, probably at Big Bang speeds at that, until it began to shrink again to a final size at or below Planck's constant. And borders create all sorts of problems, don't they? For example, it might cause a run in our string theory, and where would we be? On the run! Where to? To the opposite side of the universe from the prick, unless some other race elsewhere in the universe also popped the bubble. Can you imagine the arguments and mutual incriminations when our our survivers met theirs? "What did you crazy blighters do that for?" "US??? What do you mean us? What did YOU do it for?"
Or something. If ever physics runs out on us, cosmology is guaranteed to remain endlessly fascinating in the end.
Right?
Jon