We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive ω meson muoproduction on the proton at COMPASS using 160 GeV/c polarised μ+ and μ- beams impinging on a liquid hydrogen target. The measurement covers the range 5.0 GeV/c2< W< 17.0 GeV/c2, with the average kinematics ⟨ Q2⟩ = 2.1 (GeV/c)2, ⟨ W⟩ = 7.6 GeV/c2, and ⟨pT2⟩=0.16 (GeV/c)2. Here, Q2 denotes the virtuality of the exchanged photon, W the mass of the final hadronic system and pT the transverse momentum of the ω meson with respect to the virtual-photon direction. The measured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons (γT∗→VL) indicate a violation of s-channel helicity conservation. Additionally, we observe a sizeable contribution of unnatural-parity-exchange (UPE) transitions that decreases with increasing W. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow to evaluate in a model-dependent way the contribution of UPE transitions and assess the role of parton helicity-flip GPDs in exclusive ω production.