How can I find out why the paint by my skylight peeled?

Check the skylight itself for leaks, which may be small or may be just part of a seal.

I would be on moisture.

I found an article that might help you - https://homeguides.sfgate.com/prevent-peeling-paint-around-skylight-34244.html

Hello, Most likely condensation on the walls. There isn't proper ventilation, that's my guess. I doubt the skylight is sealed correctly.

could be heat, or latex paint over oil base paint.crape it off and use a good primer then repaint.

Probably it was lack of prep work from time of application. Was the surface clean, was there a primer and so on?

take a chip(s) of the peeling paint (large as you can get) to a paint store (big box not recommended) . the experts there should be able to examine it and tell you with 90+% certainty why the paint peeled and how to solve the problem. it is difficult to tell from your photo but it does not appear to be moisture.

Do you still have the paint can from the paint you used 2 years ago for the skylight well? If so, check whether it was latex based paint or oil based (alkyd) paint. If it's oil based paint and the underlying coat of paint was latex, the oil based paint will fail and peel off. If you're sure there is no leakage from the skylight (have you checked the walls for moisture) then scrape off the paint, sand smooth, prime with Kilz and repaint with a marine (outdoor) paint or a moisture resistant latex paint like the kind used to apply to damp basement walls. You can apply latex paint over oil (alkyd) paint, but not the other way around.

Could it be changes in temperature? I'd see how it is insulated if it isn't the paint and prep. Good luck.