We propose a new method for six-degree-of-freedom (6-DoF) autonomous camera movement for minimally invasive surgery, which, unlike previous methods, takes into account both the position and orientation information from structures in the surgical scene. In addition to locating the camera for a good view of the manipulated object, our autonomous camera takes into account workspace constraints, including the horizon and safety constraints. We developed a simulation environment to test our method on the "wire chaser" surgical training task from validated training curricula in conventional laparoscopy and robot-assisted surgery. Furthermore, we propose, for the first time, the application of the proposed autonomous camera method in video-based surgical skill assessment, an area where videos are typically recorded using fixed cameras. In a study with N=30 human subjects, we show that video examination of the autonomous camera view as it tracks the ring motion over the wire leads to more accurate user error (ring touching the wire) detection than when using a fixed camera view, or camera movement with a fixed orientation. Our preliminary work suggests that there are potential benefits to autonomous camera positioning informed by scene orientation, and this can direct designers of automated endoscopes and surgical robotic systems, especially when using chip-on-tip cameras that can be wristed for 6-DoF motion.