It is pretty simple. Skills are governed by health, stamina, magicka, and perks. Plus various other modifiers like fortify, absorb, damage, and drain.
and that's not all.
for example (just as a speculation of how it could work without attributes):
"strength" could be upgraded through skills. what i mean is, for each point gained in a skill pertaining to one handed two handed or hand to hand, the damage value of the character would go up.
1 point up in one handed skill = 1% more melee damage.
1 point up in two handed skill = 1% more melee damage.
1 point up in hand to hand skill = 1% more melee damage.
this would mean that a one handed warrior at 100 skill points would do double melee damage
if this character also specialized in two handed weapons, this warrior would do triple damage with melee attacks.
and a character that spent his time honing all his martial skills would be able to do 4x times the damage than of an inexperienced warrior. (sneak attacks did 6x times or more in oblivion so i don't think 4x damage for a master melee warrior would be overpowered)