Exp fodder is solely for leveling up the weapons. Skills are a different attribute of some weapons separate from the weapon level. If you look at the properties of a SR or SSR (and some R) weapon you will see something under skill like "Aeolus's might" with the text "medium boost to wind allies attack". The objective here is to stack like weapon skills to boost your element's attack across the board.
You can only raise the skill of a weapon by using another weapon with a skill. And as that skill gets higher, it will take more skill experience. For example, to raise the skill level of a SR weapon from 1 to 2, you either need a single SR weapon with a level 1 skill (any skill will do) or 4 R weapons with skills.
There are damage skills, health skills, crit skills, ability skills,etc. It's commonly accepted that stacking damage skills is a good way to go.
For example, here's my wind weapon pool right now. I changed the view to skills instead of attack. Note the little sword indicating attack and the level of each. I'm trying to stack as much damage up for wind as possible so all the wind characters in my party get that raw boost.
Here is the most comprehensive guide you'll find on weapon skilling.
http://gbf-english.proboards.com/thread/595/#3