Pilot Philip Garvey, 56, his wife Ann, 55, their daughter Emily, 23 and son Daniel, 20 died in the crash near Churchinford, police said.
The family from Woking, Surrey, had been flying towards Dunkeswell Aerodrome in east Devon.
Avon and Somerset Police said the six-seater plane had now been recovered from the field for further examination.
Officers are carrying out a joint investigation with the Air Accidents Investigation Branch.
The family were members of St Dunstan's Catholic Church in Woking.
Emily and Daniel were former pupils of St Dunstan's Primary School and the St John the Baptist School (SJB) in Woking.
A statement released by SJB read: "The family was highly regarded within the SJB community, Emily and Daniel were two of the nicest and kindest young people you could ever wish to meet and were extremely well liked and respected by their peers and teachers.
"Philip and Ann Garvey supported the school in every way, not only in the interest of their own children but also devoting their time supporting the school as a whole even after their children had left.
"Currently Philip Garvey was involved both in supporting our Year 13 students with interview preparation and offering current Year 11 and 12 students work experience at his company - they will be sorely missed.
"Our thoughts, prayers and deepest sympathy are with their remaining family members."
Four members of the same family who were killed in a plane crash in Somerset on Saturday have been named.