The former Manchester United star, who won two league titles and one Norwegian Cup as boss of Molde, said he was relishing his first managerial job in English football.
Mackay was dismissed after a series of disputes with Tan, leading some bookmakers to make Solskjaer odds-on to survive less than 18 months in his new job.
Solskjaer said: “I’m very confident in my own ability as a manager and I’m looking forward.
“I go into this with my eyes open and I want the fans to be proud of us when we go on the pitch.
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“I will only ask my players for 100 per cent which was hammered down to me with the gaffer (Sir Alex Ferguson) and Roy Keane at Manchester United.
“We can forgive mistakes as long as we have a go and enjoy playing in the Premier League.
“I felt I needed to be back here and I want to stay for the long, long term.”
Unfortunately for Solskjaer, it was the bookmakers who proved correct, as the Norwegian was dismissed in September after failing to prevent Cardiff’s relegation to the Championship, and making a poor start to the subsequent campaign.
He returned to Molde before later answering Manchester United’s call to replace Jose Mourinho, initially on a caretaker basis, in 2018.