And they followed that loss with a 2-1 defeat at Molineux on Saturday that knocked United out of the FA Cup at the quarter-final stage.
United are reportedly set to make Solskjaer their permanent successor to Jose Mourinho despite going into the international break with consecutive setbacks.
And Mata feels it will be important for United to respond when they return to action at home to Watford on March 30 at the start of the run-in.
"It hurts to say that Saturday's defeat against Wolves was deserved, as it saw us head out of the FA Cup and you all know the hopes that we had riding on this competition, especially after progressing through some really tough knock-out ties," Mata wrote in his regular blog.
"We fought until the very last minute but they were better than us and we have to accept it. These are the kinds of situations that put your personality and the spirit in the dressing room to the test, which is something we've talked about when we have been winning.
"I'm sure that we are going to leave this defeat behind and head towards the most important part of the season with our batteries fully charged after the international break. The time is approaching."
United's incredible comeback in Paris led to Solskjaer's side being paired with Barcelona in the draw for the Champions League quarter-finals.
It will mean a second-leg return to Camp Nou, where Solskjaer scored an injury time winner to incredibly beat Bayern Munich in the 1999 final, for United.
Mata expects a tough tie against the runaway La Liga leaders, for whom captain Lionel Messi has scored six goals in his last three appearances.
"Obviously, it is a huge challenge, but this is what this club is made of, overcoming adversity, demanding a lot from yourself and wanting to win," Mata said ahead of next month's tie.
"As Barca are one of the toughest opponents in world football, we will have to put all of that into practice.
"For the boss, returning to Camp Nou will be very special, as well as for everyone who was there on that special night in May 1999.
"This tie will ask us to put in a huge effort, but we are ready to give our best to try to go through.
"In La Liga, Barca beat Real Betis, thanks to another masterclass from Lionel Messi, and we will have to hope that he isn't so inspired when we play against them."