The double-polarization observable E was studied for the reaction gamma p -> p omega using the CEBAF Large Acceptance Spectrometer (CLAS) in Hall B at the Thomas Jefferson National Accelerator Facility and the longitudinally polarized frozen-spin target (FROST). The observable was measured from the charged decay mode of the meson, omega -> pi(+)pi(-)pi(0), using a circularly polarized tagged-photon beam with energies ranging from the. threshold at 1.1 to 2.3 GeV. A partial-wave analysis within the Bonn-Gatchina framework found dominant contributions from the 3/2(+) partial wave near threshold, which is identified with the subthreshold N(1720)3/2(+) nucleon resonance. To describe the entire data set, which consisted of omega differential cross sections and a large variety of polarization observables, further contributions from other nucleon resonances were found to be necessary. With respect to nonresonant mechanisms, pi exchange in the t channel was found to remain small across the analyzed energy range, while Pomeron t-channel exchange gradually grew from the reaction threshold to dominate all other contributions above W approximate to 2 GeV.