3D coverage path planning for UAVs is a crucial problem in diverse practical applications. However, existing methods have shown unsatisfactory system simplicity, computation efficiency, and path quality in large and complex scenes. To address these challenges, we propose FC-Planner, a skeleton-guided planning framework that can achieve fast aerial coverage of complex 3D scenes without pre-processing. We decompose the scene into several simple subspaces by a skeleton-based space decomposition (SSD). Additionally, the skeleton guides us to effortlessly determine free space. We utilize the skeleton to efficiently generate a minimal set of specialized and informative viewpoints for complete coverage. Based on SSD, a hierarchical planner effectively divides the large planning problem into independent sub-problems, enabling parallel planning for each subspace. The carefully designed global and local planning strategies are then incorporated to guarantee both high quality and efficiency in path generation. We conduct extensive benchmark and real-world tests, where FC-Planner computes over 10 times faster compared to state-of-the-art methods with shorter path and more complete coverage. The source code will be made publicly available to benefit the community. Project page: https://hkust-aerial-robotics.github.io/FC-Planner.