Airmanship is not defined in many English dictionaries however it clearly has its origins in "Seamanship" which is defined:
skill in and knowledge of the work of navigating, maintaining, and operating a vessel
Looking a little deeper:
Seamanship is the art of operating a ship or boat.
It involves a knowledge of a variety of topics and development of specialized skills including: navigation and international maritime law; weather, meteorology and forecasting; watch standing; ship-handling and small boat handling; operation of deck equipment, anchors and cables; rope work and line handling; communications; sailing; engines; execution of evolutions such as towing; cargo handling equipment, dangerous cargoes and cargo storage; dealing with emergencies; survival at sea and search and rescue; fire fighting.
Substitute aircraft for vessel or boat and you have a fairly good definition.