... I want to very strongly emphasize a few points.
1. The universe has no 'edge'. This is because we model the universe as homogeneous and isotropic. That is, there are no special places. Also, the universe at large is described by general relativity, which describes spacetime as a smooth manifold, which excludes the possibility of an edge.
So, one possibility is that the universe is infinite, it just extends forever. However, it may also be finite. If this is the case, then it's either simply or non-simply connected. That means it would be like the three-dimensional analogue of the surface of the Earth - it's definitely finite, but it has no edge. So, in a finite universe, going far enough will bring you back to your original position.
So, you can see from that why your question is flawed - there is no such thing as 'outside' of the universe because it has no edge.
2. The universe has no center, and the big bang was *not* an explosion. This will make more sense in a little bit, but it's key to remember that the universe has no center - it expands from every point in intergalactic space. The big bang was the point in time when the entire universe was filled with a hot, dense, plasma and radiation. As space underwent expansion, the radiation redshifted and cooled, and the plasma cooled into hydrogen gas. This then accumulated into clumps, which became galaxies. The space in between these galaxies continues to expand.
So, what does it mean that the universe is expanding? It means that the distance between bound objects (i.e. galaxies) is increasing. The galaxies themselves aren't moving (well, they have a very movement that has nothing to do with expansion, but this is besides the point), they just appear to be because the distance between them is getting larger. So, you can see that 'expanding' in this context is a lot different than how you think of it - the distance in between all galaxies getting larger.
...
I highly recommend reading this article, it should clear up a few misconceptions:
http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf