Big bang is not an explosion, but an expansion of space and time. Universe had almost infinite density... wait a moment. If it had ALMOST infinite density, it had a certain volume, and thus, space already existed. Am I commiting a mistake?
Krotanix : nobody really knows the answer to this. We have good evidence that the universe is expanding, and we extrapolate this back to the big bang. Then some people say the universe was once the size of a grapefruit. See for example this. Since WMAP evidence in 2013 suggested the universe is flat, some people have started saying the observable universe was once the size of a grapefruit. And that the universe was always infinite. But this is a non-sequitur, and a non-answer.
Yes, it looks as if 13.8 billion years space had a much higher density and started expanding. And it may have had a certain volume, so space may have already existed. But nobody knows for sure.
Re your follow-on question, nobody has discovered hollowness or emptiness anywhere in the centre of the universe. The expansion is thought to be like that of a raisin cake rising in an oven. There's no big hole in the middle.
You're right; space did exist. In fact, the universe was infinite in size at the moment of the Big Bang, and it still is. This is because a common misconception is wrong: The Big Bang did not happen at a point. Thinking of it like an explosion isn't correct.
You asked in a comment about the multiverse. This is speculative, to say the least. There are some theories involving objects called branes that posit that we are on the surface of some object floating through a higher dimensional "bulk". However, these theories of so-called large extra dimensions have yet to be supported by experimental evidence.