Where should I install skirt boards on my oak stairs?
You really need another skirt board for the wall. When I have my druthers, I make the skirt board the same thickness as the base that will meet it at the top and bottom so it ties in continuously. If need be, I rip a base so the top element, if it has a profile, will make it look like the base goes right up the wall along with the treads.

As a mention, the material you have for the skirtboard would be used on the outside where the wall is under the staircase, using riser brackets to help hide some of the joinery.