In oblivion we had vanity view (middle mouse button), so we can see our character from any angle. Is there a vanity view in Skyrim? Or at least can I see my character from the front in the inventory screen?

in 3rd person view u can rotate the camera around to see urself when u do not have a weapon drawn

Go in to third person, put your weapons away then use the right anologue stick or the PC equivalent to rotate toe camera.

You can go into third person view. You can rotate the camera to the front as long as your weapons are not drawn or you're not in a combat stance. You can't view your character while in the inventory.