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.
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