In terms of size, our technology is not enough to "see" the entire universe, what we know based on the Cosmic microwave background and other observations is that the universe started from a compressed state and start to expand.
This expansion gives the idea of an endless universe. The reality is that the continued expansion makes impossible to determine the end of it, due to it expanding faster than we cover the distance to see the end.
In reality we are not sure if there is any "end" or if the universe conect like in a sphere with out an "end". It would be very difficult to know it thanks to the expansion and its speed.
The Big Bang is more a theory that explains what happened after t milliseconds. Talks about this expansion, not the origin of it, but explains how the energy was concentrated and started to expand until the light was capable to escape from this mass. This first image is the cosmic microwave background.