What did the Big Bang Bang in? Did the whole of 3-D space exist around the singularity that exploded?

All of the NOVA and other physics doentaries show the Big Bang happening "in" something -- 3D space. Was empty space already there for matter to expand in, or did space [the void] enter into existence with the Big Bang? When physicists say "the universe is expanding" to they mean only that the galaxies are moving away from each other or do they mean that the volume of the 3-D space that contains all matter is getting bigger?