Silkeborg Skov came up through the youth academy at
Silkeborg, making his league debut for the club on 16 May 2013 in a 1–1 home draw with
Midtjylland. He came on at 16 years old, as an 81st minute substitute for
Jeppe Illum. He scored his first league goal for the club on 19 October 2014 in a 2–1 away defeat to FC Midtjylland. He scored in the 95th minute. Robert Skov played a key role in helping Silkeborg win the Danish 1st Division during the 2013/14 season, despite being only 17 years old. During the 2016–17 Danish Superliga season, Skov helped Silkeborg avoid relegation by becoming their leading scorer, by scoring 10 league goals from his position on the wing.
Copenhagen In January 2018, Skov completed a move to
Copenhagen. The deal cost FC Copenhagen around 7.5 million
Danish Kroner. He made his league debut for the club on 10 February 2018 in a 5–1 home victory over
Randers, playing all ninety minutes of the match. He scored his first league goal for the club a little over two weeks later, on 25 February 2018 in a 1–0 home victory over
Odense BK. He scored in the 61st minute. He scored his first club
hat-trick on 2 December 2018 in a 6–1 away victory over
Horsens. He scored in the 23rd, 66th, and 90th minutes. Not only did Robert Skov finish as top scorer in the 2018–19 season, but he broke the all time Danish Superliga record by scoring 29 league goals, that was 7 more goals than the second placed scorers in the league. The record was previously held by
Ebbe Sand who scored 28 league goals in the 1998 season. Skov was also named Danish Player of the Year for 2018. By March 2019, Skov was rumoured to
Tottenham Hotspur and
Arsenal after putting in outstanding performances in the Superliga.
TSG Hoffenheim In July 2019, Skov signed a contract with
TSG Hoffenheim for a transfer fee of €10 million. Skov scored his first goal for Hoffenheim in a 3–0
Bundesliga win over
SC Paderborn on 1 November 2019, opening the scoring with a fierce 25-yard free-kick inside two minutes. On 21 May 2024,
TSG Hoffenheim announced that he will leave the club after this season when his contract expires.
Union Berlin On 4 September 2024, Skov signed with
Union Berlin. ==International career==