Say for example I was to do Biceps on a monday and Triceps on a wednesday. Would it be more logical to do them on separate days or should each exercise be followed by it's opposite movement.

What I mean by that is if I was to do a curl, then afterwards should I do a chin press (I am not sure of it's name, but it's the movement where you lie down on your back and push the bar forward and up with your triceps). It is basically the opposite of the curl movement.

Since I would be against body part splits, then yes, I would choose antagonist pairings, which will also cause neuromuscular relaxation of the antagonist when performing an exercise, which is an advantage in interset rest periods. That's one of the advantages of a full body (possibly) or upper/lower split. On the other hand, in push/pull splits that look at movement patterns over muscle grouping, you do not the ability to use this process, but do have the advantage of synergist movements, especially for the posterior chain.

BTW, it's called a "skull crusher"

Ditto to bipennate. I favor splitting my workout into upper and lower splits as well, for the same reasons as well as the additional rest factor. Body part splits simply require too much volume in one day.