You can't. Then again, there's no need for GF's (except maybe Doomtrain, or Eden if you REALLY want to). But no, you can't skip the animations. You can't skip them in ANY FF. You CAN shorten them on other FF's though (can't remember which ones exactly).
GF damage is terrible compared to physical attacks and limit breaks once you learn to use the Junction system to strengthen your characters instead. The only exception is Eden, which is still completely inefficient since it has an animation time at about 4 times longer than the others and will only hit for a third of what your big hitters can do with their limit breaks since Eden only hits once and can't go beyond the 60k damage range.
After you learn how to play the game efficiently by junctioning properly and abusing limit breaks, the only GFs you'll have a use for are Doomtrain for Meltdown status on enemies, Cerberus for both Double and Triple status on all of your characters, Diablos in the rare case you only want to damage the enemy but not kill them, and possibly Carbuncle for the Reflect status on all party members. Junctioning 100 Tornado (available even before entering the Fire Cavern with only about 15 minutes of playing cards per 100 spells) to Strength is all you need to make a character do almost twice as much damage as a boosted GF. By the time you leave for the Field Exam, you likely won't even want the GF command on your characters if you've learned how to properly junction to stats.
Aside from the GFs listed above, all the others only do damage and do it poorly compared to what your characters can do in one turn. Even Eden only does about 4/5 of the damage a properly junctioned Squall can do with a full Renzokuken without a finisher. With the Lion Heart finisher, Eden doesn't even do 1/3 of that damage. Given summoning time, animation time, and the minimal damage, GFs just aren't efficient for damage beyond the first 20 minutes of the game. You can keeping your level low to avoid letting your enemies get levels, keeping your health in critical (yellow) throughout almost the entire game to allow for fast limits, and abusing those limits by using them over and over, you can beat the game quite easily.
By the time the Aura spell comes around on the second disc, allowing you to use limits the same way at full HP, it'll be rare for you to use something other than limits to damage bosses. --- I am a man. The CIA is just a cover.
if the animations could be skipped eden could then do 60k+ (maximum is 65535) damage and protect you from damage while charging it up which would then not require using aura in the first turn or being at low health and gives you 2 lots of health (and providing you a way of protecting yourself from 100% kill attacks wthout boring yourself watching gf animations
though quick shot renzokuken and duel are great damage dealers :)