The observable universe is constantly expanding as more light from the Big Bang reaches us. This light has been travelling for billions of years, so we are looking at the universe as it was a few billion years ago. This being said, would it be possible to look at the Big Bang in action by looking deep into the universe?
No. The furthest we can see is the cosmic microwave background radiation (CMB). Early on (after the big bang), matter was fully ionised and the electrons frequently interacted with the photons. That has two consequences. First, the radition was that of a blackbody at the same temperature as the matter. Second, the universe was opaque, i.e. photons couldn't travel very far. Due to the explosive expansion of the universe, the temperature decreased all the time until eventually atoms formed. This is called the epoch of re-combination, though the "re" makes little sense. At that point the universe became suddenly transparent and we can see most radiation emitted ever since. In particular the radiation field from the epoch of recombination has been redshifted and appears to us as the CMB.
edit To answer the speculation in the comments what if the universe was not opaque. Well, all we would be able to so is some (nearly) isotropic radiation field emitted very early on (and redshifted by the cosmic expansion since the big bang). The CMB is exactly that, but emitted at redshift $z\sim$1000 (the epoch of re-combination) rather than larger $z$.