I would NEVER exclude skull crushers from my tricep workouts, they are the "bread and butter" of all
tricep exercises...it would be like taking squats out of my leg routine...no way brother!
Perhaps it is you are doing them slightly incorrectly or your form is suffering without you even knowing it, and that is why it hurts your elbows so much? Perhaps you are hitting the triceps with skull crushers without warming up 100%? Perhaps using too much weight to get a full and controlled range of motion? It could be a number of things...so try to narrow down what is causing you pain there, instead of taking out the exercise, because it IS a great exercise.
Cable pushdowns and weighted Dips are both fantastic exercises as well, so that's awesome that you feel like they are working good for you...I would keep those in your
tricep workout arsenal.
My tricep routine generally looks like this, but the exercises change once in a while because I follow Max OT, and the routine changes every so often...here is the usual tho more often than not...
Skull crushers
cable pushdowns
Behind the neck D-bell extensions