In the second or the third movie about Bilbo, this is revealed a bit, when the Elven king is ranting against this or that. During talking, he "drops" his magic a bit, and you can see that underneath magical makeup, he's severely scarred in his face.
This is the scene, Thranduil telling Thorin Oakenshield “Do not talk to me of dragon fire! I know its wrath and ruin. I have faced the great serpents of the North!”: