I would say that my most awesome moment in Skyrim came extremely early. I mean, I've been amazed by the game in about every aspect, but the thing that truly made me appreciate the scale of Skyrim occurred right out of the starting gate, when I was deciding which Guardian Stone to take at first (the ones you come across before even reaching Riverwood).
I had just escaped Helgen with Hadvar (Imperials FTW. =P), and was walking down the mountain path to the Guardian Stones. Hadvar suggested that I pick one. Overlooking the Guardian Stones was a huge lake in front of me, with a large mountain in the background, completely devoid of clouds, majestically covered in snow. I walked up, chose my stone (Thief btw), and turned back to Hadvar, who made a comment about it never being too late to change my fate. I slowly started walking away from the stones, but then took one last, longing look at the mountain that had so captivated my gaze beforehand.
Right when I turned away from the mountain to rejoin Hadvar, I noticed a small red fox slowly trotting on the path right next to me. It stopped for a second, looked directly at me, and then continued its trek along this uphill path, disappearing into the brush right past the Guardian Stones I was standing at a moment earlier. For that one second where I watched this small creature slowly make its way across my path, I was completely enamored with the world of Skyrim. For that single moment in time, I wasn't playing a video game. I was watching a living creature make its way through the landscape.
It was at that moment that I knew I was going to enjoy every second I spent in the world of Skyrim, and so far I haven't been proven wrong.