Assigning Skill Points - Random Nature

[Synthespian]

Saving skill points won't help you in cup draw.

I've read about that in trying to find the answer to my question. And no, I'm NOT asking about the cup.

It is useful only for power training. No one needs handling (expect for GK), but we can't change that.

Yes. I'm trying to power train.

You can add skill points only on defence, attack or physical, but not on single skill.

And I know it's not assigned by specific skills, and you can only choose the categories.
But I'm asking if there's a way to get around overloading Handling (for defenders/midfielders) and crossing/aggression.
So right now, I've got a single skill point for two of my defenders.
For one of them ... if I choose Defence, it is assigning it to Handling.
Attack = Finishing
Physical = Aggression

For the other:
Defence = Handling
Attack = Crossing
Physical = Creativity.

And I don't really want some of those.
So ... is it random per gaining of a skill point? If I gain another skill for them will it change the skills chosen by the engine per category? Or is it not random and pre-defined per player?