Emerging concepts employing angular momentum of THz light for ultrafast material control rely on the measurement of undistorted intense THz fields and on the precise knowledge about sophisticated THz helicity states. Here, we establish z-cut α-quartz as a precise electro-optic THz detector for full amplitude, phase, and polarization measurement of highly intense THz fields, all at a fraction of costs of conventional THz detectors. We experimentally determine its detector response function, in excellent agreement with our modeling. Thereupon, we develop a swift and reliable protocol to precisely measure arbitrary THz polarization and helicity states. This two-dimensional electro-optic sampling in α-quartz fosters rapid and cost-efficient THz time-domain ellipsometry and enables the characterization of polarization-tailored fields for driving chiral or other helicity-sensitive quasi-particles and topologies.