Ørnskov began playing football at age 5 in the Silkeborg IF academy and came through the ranks to make well over 100 appearances in all competitions for them.

[6] On 5 July 2013, Ørnskov transferred to Danish Superliga club Brøndby IF on a three-year contract for an undisclosed fee.

[7] Brøndby-coach, Thomas Frank called the signing a "perfect fit" and lauded Ørnskov's potential as a leader.

[8] He made his debut for the club on 21 July 2013, assisting Andrew Hjulsager's goal in a match that ended in a 1–1 draw against FC Vestsjælland.

[9] He made 31 appearances in his first season back in Denmark, scoring three goals and helping Brøndby to a fourth place which secured Europa League qualifiers.

[6] The 2015–16 season saw less playing time for Ørnskov, as Christian Nørgaard emerged as a starter at his position on defensive midfield.

[13] That season he made 38 appearances in which he scored one goal as Lyngby surprisingly ended in third place and qualified for the Europa League.