If the skill occurs on the field, you can't skip the animation until you've beaten the game. If the skill occurs over a separate background, you can skip it after you've executed it a certain number of times by pressing R2.
The game is distinguishing between so called "black screen skills" that are mostly longer and have a black background and "on field skills" that take place directly on the battle field.
For black screen skills, there is a hidden counter for each one and once you have used the skill a certain number of times, the skip option will simply unlock for this skill.
In general the skip counter is higher for the first skills you can obtain and will decrease more and more for the longer and more advanced skills.
On field skills can never be skipped during the normal game, however they can, once you reach the end game content.
If skipping is possible, a new option will appear in the on-screen dialog that is visible during pattern selection.
