How do they record video of airplane flying in the air?

They use specially trained pigeons.

They are taken from another plane. There are a handful of companies that specialize in air to air filming. If it's airliner at altitude chances are very good it was taken from a Learjet, modified with a camera mount, flown by a pilot named Clay Lacy. A number of other types are used as camera platforms at lower speeds and altitudes, including a WWII B25 bomber where the tail-gunners position has been converted to an open air position for a camera man.

It's called formation flight.

Usually it is done from another plane, preferably a smaller faster one. Satellite would be to inaccurate to record a plane and would only be from above. Most internal shots on a plane are done in simulation, but often, especially in older movies, the external shots are taken in the sky.

from another plane

Clay Lacy Aviation - Astrovision camera system - from Learjet - 95% of all movies and TV commercials are made with Astrovision - Is a camera system mounted with optical system through periscope - Through top or bottom of fuselage or their Learjets - They can take pictures as high as 45,000 ft and Mach .80 - Pilot has a monitor on glareshield and "sees" what is being filmed - http://en.wikipedia.org/wiki/Clay_Lacy .