I am a secondary teacher-in-training, and to make some extra money for school, I have substitute taught at a local middle school and high school for a while now. I have no experience breaking up "big person" fights, but I have had to break up a few middle-schoolers with raging hormones and the like. I haven't used aikido or anything, but just some quick reflexes and "classroom management."
What I did was, step right in between them, almost like irimi, but with no technique attached. One kid was the very clear aggressor at this moment, so I focused on him. I maintained a stance in between the two boys, so that neither could advance against the other without touching me (neither boy was daring enough to risk the consequences of assualting a teacher, and if they did, I figured I could "take" a hit if need be). This effectively kept them seperated for the time being. While seperated, I kept my eyes focused through the aggressors eyes, and as he attempted to move past me to get back at his target, I was able to place myself in his path keeping him from doing so. Since I was able to maintain eye contact, he couldn't look at his target and in about 15 seconds he was cooled down a little bit. He was then sent to the office, and fifteen minutes later, the other boy was sent as well. Both boys were suspended, nobody was seriously hurt beyond a bruise.
Probably not very useful for breaking up bar-fights or other "big-person" fights, but smaller fights like this happen in schools all the time, and so maybe this will be useful for teachers, coaches, after-school activity planners, etc. Especially since the "breaker-upper" doesn't need to actually touch anybody and open themselves up to lawsuits. I think they key to de-escalating them is to keep the combatants seperated, and try keep them out of each other's sight until both are rational again.