Damn you for bringing this up lol.
Well, thanks!
(. . .) Or they made the UI first and decided "when is anyone going to see their face anyway?" because the only way to see that is to pan around while standing still.
But you not only see your character's face. You see it moving the head for looking things or people, and you don't need a frontal camera for that: this feature it's pretty visible with the standard third person camera view.
What you can't see in that way is —I agree—, the facial expressions of your character, like smiles, angry expressions and such.
Another theory is it was let go for optimization. Less processing required.
Actually, I don't agree with that. Back in 2006 (Oblivion), with less computing power, that could have been an excuse, but right now I seriously doubt it.
On the other hand, currently, if you hold the F key while playing Skyrim, you are able to rotate the camera for having a frontal view of your character even in middle of combat. But . . . you can't walk while doing so

Due to the difficulties involved in seeing your own character with more freedom —as in Oblivion and other games— there have been requests around here for more ways of handling the camera in Skyrim.