This paper addresses the problem of steering an Unmanned Aerial Vehicle along a given path. In the setup adopted, the vehicle is assigned a nominal path and a speed profile along it. The vehicle is then tasked to follow this nominal path independently of the temporal assignments of the mission, which is in contrast to "open-loop" trajectory tracking maneuvers. The paper builds on previous work by the authors on path-following control and derives a new control algorithm that uses the Special Orthogonal group SO(3) in the formulation of the attitude control problem. This formulation avoids the geometric singularities and complexities that appear when dealing with local parameterizations of the vehicle's attitude, and leads thus to a singularity-free path-following control law. Flight test results performed in Camp Roberts, CA, demonstrate the efficacy of the path-following control algorithm developed in this paper.