Is there a way to pick up arrows that have been evaded by my player?
Arrows shot by a skeleton or stray cannot be picked up.
With the exception of mods for the PC version, its not possible in any version of the game including PE. This is likely due to the fact that it would make it pretty easy to farm arrows form a skeleton.
No. Arrows shot by skeletons cannot be picked up by the player. Only arrows shot by the player (without the infinity enchantment) and missed can be picked up.