I'm having this same issue. I get an error when I try changing my skin on the official launcher, the Minecraft.net website, and 3rd party launchers. Also, if this information is helpful at all, I get an "AccessDenied" error when trying to open s3.amazonaws.com/MinecraftSkins
So I found a "fix". I was able to change my skin by logging onto minecraft.net with my iPhone on mobile data. So I'm wondering if this issue is about internet providers blocking the skin servers.