The simple "no cost" option. The camera operator sits in the back and films from the shoulder through an open door or window. This is the standard choice for news gathering and can also be very effective for standard wide lens landscapes when you just haven't got the budget for a camera mount. There are also occasions such as filming in remote locations where you simply can't get a camera mount and hand held is the only option.
If you are filming from the shoulder for whatever reason, just don't try to be too ambitious. Try and use a wide focal length lens, pick a calm day if you can and you're likely to achieve a good result. Trying to hold a steady long focal length close up from 500 feet away isn't really achievable and the results will be very wobbly and probably unuseable.
On a point of safety: Make sure that the camerman is securely strapped in as per the pilot's instructions and that the camera is attached separately to the aircraft. All accessories such as batteries, monitors, lenses etc should also be held in bags or cases which are tied to the aircraft. This is probably required by your production insurance.
Top Tips:
It can get very windy if you're moving at over 60 knots with the door off and things can easily get blown off!
Double knot your cameraman's shoe laces .
Do not use a matte box on the camera.
Make sure the lens is properly secured to the camera.
Make sure that the lens hood is tightly fitted and think about taping it on as well.
Carry spare tape or film, batteries, and lens cleaner.
Don't wear a hat.