The big bang did not happen in a single point in space. Why? Because there was no space there for it to be in! The big bang was an expansion OF space itself. An expansion OF the universe itself as a whole. Every point in space is expanding away from every other point in space. There is no center, and there is no edge.
Now, a common example is the balloon. Imagie putting a bunch of dots, equally spaced, with a marker on an uninflated baloon. When you blow into it, the balloon expands. Every point on that balloon is moving away from every other point on the balloon. A dot that is 10 dots away from another dot is moving away much faster than one that is only 5 dots from that same dot.
Now, the key here is to realize that while the balloon is expanding INTO space, the expansion OF space itself does NOT require it to be expanding INTO something. I know it doesn't make any sense, but there is good reason that it is described that way.