Could we say that the Big Bang is still happening? I mean, isn't the observed expansion of the Universe the Big Bang in action? In other words, if we draw planets, stars, galaxies, star systems on a piece of paper and then crush the sheet into a ball and leave it, it will slowly begin to unfold and this will happen for a while. If we are patient enough we can see the sheet of paper unfolded again, as it was in the first moment.
Ever since the big bang happened our universe has been expanding. As a matter of fact, the continued large-scale expansion of the universe is one of our best arguments for why we know that our universe started from a very small dense state.
But while we continue to see other galaxies flying away from us faster and faster our own milky way galaxy and all the stars, planets, humans and paper balls inside of it are bound by local forces that are far stronger. Our stars and planets hold together by gravity, so while the universe at large will continue to grow on large scales our solar system will stay at the same size. Similarly, humans and paper are held together by electromagnetic forces. So to answer your question: a piece of paper on earth does not expand over time, to see the effects of the big bang we need to observe faraway galaxies.
Thanks for your great question, the expansion of the universe is a fascinating thing to think about since it is so different from our everyday life.