Skinner's second goal came halfway through the middle period for a 2-0 lead and was the first of three quick Carolina scores within a 4-minute span.
Additional goals from Andrej Sekera and Jiri Tlusty followed in short order as the Hurricanes took the first contest of a two-game road trip.
Justin Peters stopped 26-of-27 shots to earn the victory.
Mike Green got his first goal of the season for the Capitals.
Braden Holtby allowed four scores on 23 shots through two periods before yielding to Philipp Grubauer, who stopped 9-of-9 shots the rest of the way.