The move was announced by Boston College on Tuesday.
Donahue has spent the past 10 seasons at Cornell and led the Big Red to three consecutive NCAA tournament appearances - including a Sweet 16 berth this past season.
"Anytime somebody leaves like that, it's hard to swallow," Cornell guard Louis Dale said in Ithaca, N.Y., where Donahue flew back on Tuesday so he could meet with his team on campus. "But you, at the same time, were such a fan of coach Donahue that we're BC fans now. We want to see him do well.
"That just shows what kind of coach he was and a great person. Just because he's leaving this institution doesn't mean that we're going to turn our backs on him."
Skinner was 247-165 in 13 seasons and went to seven NCAA tournaments in the past 10 years, but the Eagles hadn't gone to the Big Dance in two of the last three seasons and one of the primary reasons was the departure of former assistants Bill Coen and Ed Cooley.
Coen and Cooley took jobs at Northeastern and Fairfield, respectively - and both were interviewed by BC athletic director Gene DeFilippo.
The Associated Press contributed to this report.