In this paper, a novel two-dimensional super-resolution angle-of-departure (AoD) and angle-of-arrival (AoA) estimation technique is proposed for wideband millimeter-wave multiple-input multiple-output systems with cross-polarized antenna elements. The key ingredient of the proposed method is to form custom designed beam pairs, and devise an invertible function of the AoD/AoA to be estimated from the corresponding beam pairs. Further, a new multi-layer reference signal structure is developed for the proposed method to facilitate angle estimation for wideband channels with cross-polarized antenna elements. To facilitate feedback in closed-loop frequency division duplexing systems, a novel differential feedback strategy is proposed aiming at feedback reduction for the two-dimensional angle estimation. Numerical results demonstrate that by using the proposed method, good azimuth/elevation AoD and AoA estimation performance can be achieved under different levels of signal-to-noise ratio, channel conditions, and antenna array configurations.