The photoproduction of $\omega$ mesons off the proton has been studied in the
reaction $\gamma p\to p\,\omega$ 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 $\omega\to\pi^+\pi^-\pi^0$,
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, $\Sigma$, are also presented, which
agree well with previous CLAS results and extend the world database up to 2.1
GeV. These data and additional $\omega$-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.