Mackay parted company with the Welsh outfit in December amid reports of a falling-out with owner Vincent Tan, and Solskaer subsequently took the reins at the Cardiff City Stadium.
One of the disagreements was thought to revolve around the club's business in the transfer market, with Cardiff releasing a statement on Thursday where chief executive Simon Lim claimed Mackay and former head of recruitment Iain Moody were guilty of "imprudent and careless management".
The deal to bring Andreas Cornelius from FC Copenhagen for a club-record fee was pinpointed as a particular failure, with the striker returning to his former club last month after failing to score in 10 appearances for Cardiff.
The League Managers' Association then waded into to the issue by defending Mackay, claiming Lim's words were "inappropriate and unprofessional".
While criticising Mackay's movements in the transfer market, Lim praised the work done by Solskjaer in the January window, but the Norwegian refused to offer an opinion on his predecessor's situation.
"It is early doors for me and it is encouraging they [the board] are happy with what I have done but what happened in the past is not for me," he said.
"It guides what resources they give me and what remit I have. I'm just running the club as I would like to.
"It's not got anything to do with me, what happened before. I am just focused on my team because the January window was chaotic."
Cardiff continue their season with an FA Cup fifth-round tie against holders Wigan Athletic on Saturday.
After winning the competition last season, Wigan went on to be relegated from the Premier League - and Solskjaer is determined to avoid a similar fate.
"They proved last season what the FA Cup was all about, a chance for smaller clubs to win trophies," he added.
"I would like to go one better than they did and win the cup and stay up, but in any way, shape or form, if you play in a cup game you want to win it.
"It's a chance for some players to come in and show they should be playing in the league and I do not go into any game hoping we lose."