Not necessarily …. pH depends mostly on the levels of CO2 inside your home. You can run an outside airline or employ a CO2 scrubber, but if your tank is 'marinating' in a high CO2 room then neither of those things are going to make much of a difference. If your tank is poorly circulated, or has less optimal dimensions, it is certainly possible that the skimmer will help with oxygenation. In my systems, neither is true and skimmer makes very little difference to oxygen levels (at least, it didn't when i used to measure the levels).
I do run a skimmer, and have no plans to remove it; I do think it helps to remove organics and particulates. I also run the output of my CaRx into the skimmer so it can blow off any excess CO2.