Solskjaer took charge on an interim basis in December when Jose Mourinho was sacked, the former United striker quickly inspiring an upturn in form.
Having left his post at Molde to return to Old Trafford, United won its first eight games in a row in all competitions under Solskjaer's steady hand.
Restoring the feel-good factor at the club also helped Solskjaer to inspire an amazing comeback in United's Champions League last-16 tie against Paris Saint-Germain.
Despite losing the first leg 2-0 at home, a late Marcus Rashford penalty awarded following a VAR check sent United improbably through to the quarter-finals, where they will face Barcelona.
"That's the demand at United, the success," Solskjaer said in an interview with United's official app. "Striving to improve, winning trophies, that's another challenge for us and we want to be up there challenging every season from now on.
"I wouldn't have said yes if I wasn't looking forward to it and if I didn't believe in the club and in the potential in what we're doing. I feel and I hope I can do this job justice. That's the first thing I asked myself when I got the call a few months ago and hopefully now I can even improve on it.
"Very excited, of course. Every day is a challenge and a test. Of course we've got Watford on Saturday, we've got Wolves, they're the best two of the rest this season, seventh and eighth, so short-term that's my priority.
"Then games coming up and the summer transfer window, improving the squad and talking to players about how we want to approach next season.
"Obviously I've been discussing with [executive vice-chairman] Ed [Woodward] and the club, my ideas for the future - now more or less it is my decisions to make.
"I'm looking forward to it. If you don't look forward to working for this club then that's wrong. And I can't wait.
"We've spoken about it before but I've prepared, more or less, my whole adult life ever since I got my injury to be a manager somewhere. I've always been stupid or naive enough to dream about this job."