Precise measurement and characterization of millimeter wave channels requires antennas capable of high angular resolution to resolve closely spaced multipath sources. To achieve angular resolution on the order of a few degrees these antennas must be electrically large which is impractical for phased array architectures at millimeter wave frequencies. An alternative approach is to synthesize a virtual aperture in space by using an accurate mechanical positioner to move a receive antenna to points along a sampling grid. An advantage of creating virtual apertures is that the received signal is digitized at every spatial sample position which enables the use of sophisticated angle estimation algorithms such as maximum likelihood (ML) techniques. The main contribution of this paper is a new gradient-based implementation of maximum likelihood angle estimation that was demonstrated on virtual array data collected at 28 GHz using a vector network analyzer (VNA).