Skuse, who has spent his entire nine-year career with the Robins, will leave the recently relegated side along with striker Jon Stead, who plans to move back north, Mark Wilson, Brian Howard and Matthew Bates, whose short-term deals have expired, and goalkeepers Dean Gerken and Lewis Carey.
Director of football Keith Burt said: "We wish all of the players who are leaving the football club well for the future.
"With Cole, he's been an exemplary servant for this football club and we fully understand his reasons for moving on at this time in his career.
"For Jon the lure of playing back up north has been too strong and we wish him luck for whatever his future career holds."
City also confirmed their intention to extend the contracts of Tom Heaton, Lewin Nyatanga and Louis Carey.
"The head coach believes Tom and Lewin would be key players for us in League One next season and we'll be making offers to them both," Burt added.
"With Louis, his experience and know-how is seen as important for what we believe will be a young squad."