After starting his career with the local club Stabæk, Skjelvik moved to Sweden where he played in Allsvenskan with Helsingborg and Kalmar, before he returned to Norway and joined Rosenborg in July 2013.

He has represented Norway from under-18 to under-21 level, but was left out of the Norwegian squad for the 2013 UEFA European Under-21 Football Championship due to an injury.

[5] Skjelvik made his debut on 13 August, coming on as a substitution in the 77th minute and also managed to score a goal in the 3–1 win against local rivals Trelleborgs FF.

[citation needed] After winning both the Allsvenskan and the Svenska Cupen with Helsingborg in 2011, Skjelvik moved to Kalmar FF ahead of the 2012 season and signed a three-year contract with the club.

[8] In November 2017, Skjelvik decided not to extend his contract with Rosenborg and would leave as a free agent at the end of the 2017 season.