Three tiers of skills:
You are UNSKILLED with Skills outside your class list or not chosen, and take a -2 to them, and cannot take 10 or 20 on them.
You are Familiar with all Skills on your Class list, and may use them with no bonus or penalty. A Jack of All Trades would explicitly be able to take 10 or 20 with them as if they were Skilled.
You are Skilled and Get your proficiency bonus with any Skills you chose to be proficient in.
Where stat bonuses would fit into this would be either proficient or Jack of all Trades.
--- Then all you need to do is make the Class List of Skills, and you are good. So you can actually have a Wizard who can roll for all the Lore/Knowledge skills, for instance, even if he's not good at them.