We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive $omega$ meson muoproduction on the proton at COMPASS using 160 GeV/$c$ polarised $ mu ^{+}$ and $ mu ^{-}$ beams impinging on a liquid hydrogen target. The measurement covers the range 5.0 GeV/$c^2$ $< W <$ 17.0 GeV/$c^2$, with the average kinematics $langle Q^{2} rangle=$ 2.1 (GeV/$c$)$^2$, $langle W rangle= 7.6$ GeV/$c^2$, and $langle p^{2}_{rm T} rangle = 0.16$ (GeV/$c$)$^2$. Here, $Q^2$ denotes the virtuality of the exchanged photon, $W$ the mass of the final hadronic system and $p_T$ the transverse momentum of the $omega$ 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 ($gamma^*_T to V_L$) 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 $omega$ production.
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