The body of Graham Connell, 31, was discovered in a gully in the Jacob's Ladder area on Monday.
It has also emerged his five companions, who were left disorientated and without a map after Mr Connell's fatal accident, were less than an hour from reaching safety when they took a wrong turn and ended stranded on a mountain overnight.
Tributes have been paid to Mr Connell, of Castleford, West Yorkshire, who was one of a group of six climbers from the Leeds University Hiking Club who ventured to the Northern Corries area on Sunday.
A report on the incident is to be sent the procurator- fiscal, but it has emerged Mr Connell had been leading a party of climbers from the university and had gone to help one who had suffered a fall, only to fall more than 300ft himself at the gully in Coire an t-Sneachda, a popular climbing area.
Students who were involved or have been affected by the tragedy have been offered counselling.
Edmund Elliott, of Leeds University, said: "I'd personally like to acknowledge the death of a lifetime and hugely committed member of both the Hiking Society and the Union, Graham Connell. Of course my deepest sympathy goes out to his family and friends at this difficult time."
Around 35 of the club's members had travelled to the Cairngorms on Friday night for a weekend of climbing.
Initially a group of seven headed out on Sunday, but one was airlifted as the expedition unravelled after running into difficulty.
After Mr Connell's fall, the five set out to find help but had not realised they were very close to the Cairngorm ski slopes, where they could have raised the alarm.
One member of the Cairngorm Mountain Rescue Team said the climbers were no more than 40 minutes away. However, they turned and walked miles through the night in a south-easterly direction unable to secure a phone signal until 10am on Sunday, when they contacted Grampian police.
They were eventually located at Carn Tarsuinn, miles from the mountain rescue team search.
One climber, who did not want to be named, said "I think it is fair to say that, if you have a party of seven out in the hills, there really should be more than one person with a map, because it is so easy to get split up when bad weather comes in."
Bob Kinnaird, principal of Glenmore Lodge, Scotland's National Outdoor Training Centre on the foothills of the Cairngorms, said: "The forecast for the Sunday was for stronger winds, and with precipitation, snow, coming in later .
"In the morning it looked better, but some people were surprised by how stormy it got later on. So people were becoming disorientated."
He said the impact of wind on temperatures could also catch people out. "Winds for that day were predicted to be 30mph to 40mph, so that could have meant temperatures dropping the equivalent of -26˚C. It rips the heat out of you, so you can quickly get into very serious trouble."
A spokesman for Leeds University said Mr Connell had completed a physics programme in 2004, graduating with a diploma in higher education.
He said:"We are sorry to learn of the death of one of our former students and our thoughts are with his family and friends.
"We are working with Leeds University Union and our counselling services to offer support to students who have been affected by this tragedy."