Does Skirmish apply to both attacking and defending?
No, Skirmish is only active on attacking

You can only Fight on your turn. This means that Skirmish is only active when you select that creature to Fight as an action, negating any damage you would be dealt. 
If a creature with skirmish is targeted with a Fight action by an opponent, it will still take damage.

From the official rules:


  Any ready creature of the active house may fight. When a creature
  is used to fight, its controller chooses one eligible creature controlled
  by the opponent as the target of the attack. Each of the two creatures
  deals an amount of damage equal to its power (the value to the left
  of the card’s title) to the other creature