Exclusive electroproduction of \(\omega\) mesons on unpolarized hydrogen and deuterium targets is studied in the kinematic region of Q\(^2\)>1.0 GeV\(^2\), 3.0 GeV < W < 6.3 GeV, and -t'< 0.2 GeV\(^2\). Results on the angular distribution of the \(\omega\) meson, including its decay products, are presented. The data were accumulated with the HERMES forward spectrometer during the 1996-2007 running period using the 27.6 GeV longitudinally polarized electron or positron beam of HERA. The determination of the virtual-photon longitudinal-to-transverse cross-section ratio reveals that a considerable part of the cross section arises from transversely polarized photons. Spin density matrix elements are presented in projections of Q\(^2\) or -t'. Violation of s-channel helicity conservation is observed for some of these elements. A sizable contribution from unnatural-parity-exchange amplitudes is found and the phase shift between those amplitudes that describe transverse \(\omega\) production by longitudinal and transverse virtual photons, \(\gamma^{*}_{L} \to \omega_{T}\) and \(\gamma^{*}_{T} \to \omega_{T}\), is determined for the first time. A hierarchy of helicity amplitudes is established, which mainly means that the unnatural-parity-exchange amplitude describing the \(\gamma^*_T \to \omega_T\) transition dominates over the two natural-parity-exchange amplitudes describing the \(\gamma^*_L \to \omega_L\) and \(\gamma^*_T \to \omega_T\) transitions, with the latter two being of similar magnitude. Good agreement is found between the HERMES proton data and results of a pQCD-inspired phenomenological model that includes pion-pole contributions, which are of unnatural parity.