The photoproduction of ω mesons off the proton has been studied in the reaction γp→pω using the CEBAF Large Acceptance Spectrometer (CLAS) and the frozen-spin target (FROST) in Hall B at the Thomas Jefferson National Accelerator Facility. For the first time, the target asymmetry, T, has been measured in photoproduction from the decay ω→π+π−π0, using a transversely-polarized target with energies ranging from just above the reaction threshold up to 2.8 GeV. Significant non-zero values are observed for these asymmetries, reaching about 30-40% in the third-resonance region. New measurements for the photon-beam asymmetry, Σ, are also presented, which agree well with previous CLAS results and extend the world database up to 2.1 GeV. These data and additional ω-photoproduction observables from CLAS were included in a partial-wave analysis within the Bonn-Gatchina framework. Significant contributions from s-channel resonance production were found in addition to t-channel exchange processes.