Skjelvik played for Rosenborg BK since 2013, where he made 163 appearances in all competitions, including six goals and 13 assists from his position in central defense. While with the club, he led his side to three first-place finishes and two second-place finishes in the Eliteserien table. Additionally, he appeared in 11 Europa League matches in total during his time there.
Prior to playing for Rosenborg, the Norwegian represented Kalmar (Sweden) and Stabæk (Norway) during his time as a professional. In all, he has made 240 appearances in the top-flight divisions in Norway and Sweden, scoring nine goals and adding 23 assists.
Internationally, Skjelvik has represented Norway in seven games during his career – including three FIFA World Cup Qualifying games and three international friendlies – making his debut for his country in 2013 at the age of 22 in a friendly against Denmark. He was a part of the team’s recent campaign in 2018 FIFA World Cup Qualifying games, going 90 minutes against Azerbaijan and Germany, helping Norway finish with 15 points in Group C.