The differential cross sections and unpolarized spin-density matrix elements for the reaction $gamma pto pomega$ were measured using the CBELSA/TAPS experiment for initial photon energies ranging from the reaction threshold to 2.5 GeV. These observables were measured from the radiative decay of the $omega$ meson, $omegatopi^0gamma$. The cross sections cover the full angular range and show the full extent of the $t$-channel forward rise. The overall shape of the angular distributions in the differential cross sections and unpolarized spin-density matrix elements are in fair agreement with previous data. In addition, for the first time, a beam of linearly-polarized tagged photons in the energy range from 1150 MeV to 1650 MeV was used to extract polarized spin-density matrix elements. These data were included in the Bonn-Gatchina partial wave analysis (PWA). The dominant contribution to $omega$ photoproduction near threshold was found to be the $3/2^+$ partial wave, which is primarily due to the sub-threshold $N(1720),3/2^+$ resonance. At higher energies, pomeron-exchange was found to dominate whereas $pi$-exchange remained small. These $t$-channel contributions as well as further contributions from nucleon resonances were necessary to describe the entire dataset: the $1/2^-$, $3/2^-$, and $5/2^+$ partial waves were also found to contribute significantly.
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