Skelton resigned immediately after the 1-1 draw at Stafford Rangers, only his second game in charge, much to the surprise of the club's board and fans.
In a statement, the club said: "His decision came as a shock to the Board of directors, who felt they had appointed the right man to take the club forward. Chairman, John Mackay, worked tirelessly over the Christmas period to negotiate his release from Carlisle United, and was understandably disappointed by the news.
"The Board had structured a full-time post for Skelton, which would have seen him combine his managerial duties with the community side of the club.
"The directors will review the situation over the next few days and remain grateful to Lee Andrews as he continues to lead the first team as he has throughout what has been a difficult period for all concerned."
Workington sit second from bottom in the EVO-STIK League Premier Division and travel to Gainsborough Trinity on Saturday.