Pilot Philip Garvey, 56, his wife Ann, 55, their daughter Emily, 23, and son Daniel, 20, died in the incident near Churchinford in the Blackdown Hills south of Taunton, Somerset.
It's unclear what happened to their six seater plane.
A spokesman for Avon and Somerset Police said: "Our thoughts are with [the family] at this extremely difficult time.
"A joint investigation is under way with the Air Accidents Investigation Branch (AAIB).
"The focus of the activity will be investigative inquiries and then recovery of the aircraft."
Police earlier said that 27 fire and rescue personnel plus specialist vehicles were sent to the scene, along with ambulances, the local air ambulance and a police helicopter.
The family's priest at St Dunstan's Church, Woking, Francis Harrington has been paying tribute.
He told Premier: "It's a tragedy. A huge tragedy. The whole area and the whole parish is devastated.
"Nobody can get their heads around it. They were a lovely family and were highly respected. What can we say? Why do these things happen? I don't know. We will have to wait for the official report.
"Philip was very involved with the music at mass, where he played the guitar. I have known him for around 20 years and I have seen the family grow up. They were a lovely family. Sociable, pleasant, very nice. They will be sorely missed."