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Unmanned Aerial Vehicles (UAVs) often lack the size, weight, and power to support large antenna arrays or a large number of radio chains. Despite such limitations, emerging applications that require the use of swarms, where UAVs form a pattern and coordinate towards a common goal, must have the capability to transmit in any direction in three-dimensional (3D) space from moment to moment. In this work, we design a measurement study to evaluate the role of antenna polarization diversity on UAV systems communicating in arbitrary 3D space. To do so, we construct flight patterns where one transmitting UAV is hovering at a high altitude (80 m) and a receiving UAV hovers at 114 different positions that span 3D space at a radial distance of approximately 20 m along equally-spaced elevation and azimuth angles. To understand the role of diverse antenna polarizations, both UAVs have a horizontally-mounted antenna and a vertically-mounted antenna-each attached to a dedicated radio chain-creating four wireless channels. With this measurement campaign, we seek to understand how to optimally select an antenna orientation and quantify the gains in such selections.