The Norwegian was sacked by the Red Devils at the weekend after a 4-1 defeat to Watford and admits that he fulfilled a dream by managing the Premier League side.
Solskjaer told United's website: "First of all, I want to thank the board and the owners for giving me the opportunity because it's not for everyone and I've had the opportunity.
"I'm so honoured and privileged to have been trusted to take the club forward, and I really hope that I leave it in a better state than when I came.
"It's one of those things you dream of in your life. When you've been a player, when you've been a reserve-team coach, the next job then, the only dream and the only thing you haven't done is to manage the club, and I have now."
Solskjaer had come under increasing pressure this season following humiliating losses to both Liverpool and Manchester City and expressed disappointment that he was unable to lead the club to silverware during his three years in charge.
He said: "I wanted us to take the next step to challenge for the league, to win trophies, but unfortunately I couldn't get the results we needed and it's time for me to step aside.
"Second place last season ahead of arguably one of the best teams in Europe (Liverpool), I think that's a great achievement as well.
"And we were so close in Europe. Sometimes that's the fine margin for you, one penalty."