The GlueX experiment at Jefferson Lab aims to study the light meson spectrum with an emphasis on the search for hybrid mesons. To this end, a linearly-polarized $9\,$GeV photon beam impinges on a hydrogen target contained within a hermetic detector with near-complete neutral and charged particle coverage. In 2018, the experiment completed its first phase of data taking in its design configuration and the analyzed data already exceeds previous experiments for polarized photoproduction in this energy regime by orders of magnitude. Polarization observables like spin-density matrix elements provide valuable input for the theoretical description of the production mechanism, which will be essential for the interpretation of possible exotic meson signals. We will present results for the photoproduction of vector mesons, focusing on the unprecedented statistical precision of the spin-density matrix elements for the $\rho$(770) meson.