At the moment I know you are waiting for Joe’s reply but since I saw the question I will my give you my 2 cents and hope it helps lol. This could be better explained by pictures but Joe’s principlies around exercise selection are 3.
ALIGNMENT – which is the reason why you are having pain in the elbows
So first you should know that the elbow is a hinge joint meaning it only moves in one direction up and down. Now gravity as we know only pulls us down. So its a straight line of force downwards. When you are doing skull crushers specially with a barbell your arms tend to flare up sideways. So the elbows are no longer in the same direction as the line of force (gravity) which is down. Its kinda hard for me to explain this without images so I hope Joe’s answer will be a lot more clear than mine.