You are imagining the big bang as an explosion of matter expanding into empty space.

The theory is that when the big bang occurred there was no space and there was no time.  As there was no time the initial expansion took place in an instance and it was only at this point that time existed and therefore speed existed.

Whilst nothing can travel faster than the speed of light there is nothing in any of Einsteins various theories that says the universe can't expand faster than the speed of light.