Flt Lt Jon Egging, 33, from Rutland, died when his Hawk T1 aircraft crashed near Bournemouth Airport after a display on Saturday afternoon.
Witnesses described seeing his plane hit the ground in a field by the River Stour near the village of Throop.
Books of condolence are to be opened in Bournemouth later for public tributes.
Dorset Police said Flt Lt Egging, known as 'Eggman', was thrown from the aircraft and was pronounced dead at the scene.
Flt Lt Egging flew on the right hand outside of the Diamond Nine formation
Shortly before the crash, his wife, Dr Emma Egging, had watched him take part in a display at the Bournemouth Air Festival.
She said later: "Watching him today, I was the proudest I've ever been. I loved everything about him, and he will be missed."
Gp Capt Simon Blake, the commandant of the RAF Central Flying School, said the Ministry of Defence had begun an investigation into the cause of the crash and all Red Arrows Hawk T1 aircraft, based at RAF Scampton in Lincolnshire, had been grounded as a standard safety measure.
He said: "At this juncture it is too early to speculate as to the cause of the accident.
"In accordance with normal policy the Ministry of Defence has appointed a service inquiry to investigate the circumstances surrounding the accident and... have grounded the Hawk T1 temporarily until its safety can be assured.
"As for the rest of the season it is too early to speculate as to when the Red Arrows will be back on the public circuit but suffice to say for the short term they will not be able to perform in public."
The crash site was cordoned off by police soon after the incident and only people living inside the zone were being allowed access.
The Military Air Accident Investigation Branch has started an inquiry.
Flt Lt Egging's aircraft was one of nine Red Arrows planes to take part in the display over the seafront.
The plane plunged into a field near the River Stour
The RAF said he joined the team in the autumn of 2010 and flew on the right hand outside of the famous Diamond Nine formation.
Nicholas Gore, 22, was walking with a friend close to the river when he saw all nine Red Arrows overhead.
"There were quite a few people watching and we saw them go over but one seemed quite low," he said.
"They then disappeared behind trees and I heard a crack, not an explosion, just a crack and we got further down and I saw the plane with its red tail in the air and its nose in the river."
Another witness, who did not want to be named, said the plane had skidded for several hundred metres along the riverbank after it crashed.