Endowing robots with the ability of skill learning enables them to be versatile and skillful in performing various tasks. This paper proposes a neuro-fuzzy-based, self-organizing skill-learning framework, which differs from previous work in its capability of decomposing a skill by self-categorizing it into significant stimulus-response units (SRU, a fundamental unit of our skill representation), and self-organizing learned skills into a new skill. The proposed neuro-fuzzy-based, self-organizing skill-learning framework can be realized by skill decomposition and skill synthesis. Skill decomposition aims at representing a skill and acquiring it by SRUs, and is implemented by stages with a five-layer neuro-fuzzy network with supervised learning, resolution control, and reinforcement learning to enable robots to identify a sufficient number of significant SRUs for accomplishing a given task without extraneous actions. Skill synthesis aims at organizing a new skill by sequentially planning learned skills composed of SRUs, and is realized by stages, which establish common SRUs between two similar skills and self-organize a new skill from these common SRUs and additional new SRUs by reinforcement learning. Computer simulations and experiments with a Pioneer 3-DX mobile robot were conducted to validate the self-organizing capability of the proposed skill-learning framework in identifying significant SRUs from task examples and in common SRUs between similar skills and learning new skills from learned skills.