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Spin density matrix elements (SDMEs) have been determined for exclusive $\omega$ meson production on unpolarized protons, in the COMPASS kinematic region of 1.0 (GeV/$c^2$) $< Q^{2} <$ 10.0 (GeV/$c^2$), 5.0 GeV/$c^2$ $< W < $ 17.0 GeV/$c^2$ and 0.01 (GeV/$c^2$) $
= 2.1$~(GeV/$c^2$), $\left< W \right>= 7.6$~GeV/$c^2$ and $\left
= 0.16$~(GeV/$c^2$) mean values. The analysis is based on data taken in 2012 with the 160 GeV muon beam and the liquid hydrogen target. Using extracted SDMEs the contribution of unnatural parity exchange amplitudes, the longitudinal-to-transverse cross section ratio, and the hypothesis of s-channel helicity conservation (SCHC) were studied. A sizable contribution of unnatural parity exchange amplitudes is found for the exclusive $\omega$ meson muoproduction, although there is a clear indication of its decrease with increasing W. Certain matrix elements e.g. $ r^{5}_{00}$ corresponding to transition $\gamma^*_T \to V_L$ indicate violation of SCHC in exclusive $\omega $ production.