The first measurements of the beam-target-helicity-asymmetries \(E\) and \(G\) in the photoproduction of \(\omega\)-mesons off protons at the CBELSA/TAPS experiment are reported. \(E\) (\(G\)) was measured using circularly (linearly) polarised photons and a longitudinally polarised target. \(E\) was measured over the photon energy range from close to threshold (\(E_\gamma = 1108\)~MeV) to \(E_\gamma = 2300\)~MeV and \(G\) at a single energy interval of \(1108 < E_\gamma <1300\)~MeV. Both measurements cover the full solid angle. The observables \(E\) and \(G\) are highly sensitive to the contribution of baryon resonances, with \(E\) acting as a helicity filter in the \(s\)-channel. The new results indicate significant \(s\)-channel resonance contributions together with contributions from \(t\)-channel exchange processes. A partial wave analysis reveals strong contributions from the partial waves with spin-parity \(J^P=3/2^+, 5/2^+\), and \(3/2^-\).