Amaterasu burns until there's nothing there.But couldn't he have just skinned his arm?
Well saidIt was spreading up his arm. His reaction time was quite fast much more time would have caused him to lose more. If he skinned his arm he would have the problem of most likely getting some on other body parts. The only sure thing was to remove the entire part where the fire was. Also he was in a battle there is no time to do slow moves like skinning he needed to do this asap.