There are several instances of this in the game already, but they are all part of quests.

At Wyrmrest Temple in Dragonblight, there are several quests, one a daily, where you ride a dragon and shoot down other dragons while in the air.

In Howling Fjord, at Steel Gate, there is another daily where you fly a plane with a machine gun and rocket launcher, to defend the dig site against gargoyles.

There's another dragon daily at the top of the nexus (need to fly to get there)

And lastly, Storm Peaks has a few quests for the Sons of Hodir chain where you fight vyrkul and other dragons in midair, with your own class abilities.

I think more aerial fights were planned for Wintergrasp, but were removed due to balance issues.

I believe that was in beta, but was yanked at the last minute because they couldn't figure out how to balance it. Apparently airplanes were extremely overpowered in Wintergrasp, which is where they were used, because they could just gun down people below them and couldn't die.

Howling Fjord has dog fights in the gnome airplanes vs Gargoyles, a daily quest.