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OK, that is going to depend on what you mean by space. There are several definitions which are all internally consistent mathematically. However, each one describes space differently. We do not have enough data to sort the situation out fully and some versions are of a nature that we can never possibly hope to sort them out.

Bearing that in mind, there are a few ways in which space could have existed before the big bang. For one point, it sounds like you are trying to get to a different idea altogether, specifically what went bang in the first place and where was it located?

A few theories do support the idea that the universe is embedded in some other “place” but that we could never escape to that place even with anything unmanned to take measurements. However, if we stretch the idea to say that we did just that, we could look back and see our universe as an expanding but finite four dimensional sphere. That would be the view from the outside but back at home, the view from the inside might still be of infinite space time. The thing is that there is no compelling reason for the universe to be infinite when viewed from the inside but at our current state of understanding, we have not ruled that out either.

Now depending on what the actual case is for an outside stage where the universe is, there may be another way to do a similar experiment if we could build a big enough super collider.

There is one model of that outside place where it could be described sort of like a mountain range (the theory actually uses the term landscape but I think mountain range might offer a clearer mental picture). Anyway the idea is that there would be plateaus, ledges and valleys each at a different basic “height” and those are the places where universes can rest in a stable condition. For each of those places, the various constants such as the strength of gravity and the speed of light would be different from what we have in our universe.

So, with a powerful enough machine, we might be able to bridge the gap from one place to another and we would see a baby universe form in the machine. In fact, there is a sense in which we may already be doing just that. The RHIC can be run in such a way that it makes tiny black holes that evaporate quickly. So in that sense, our universe could be the stage in which other universes are built. That is, of course, only if the outside place is really like that.