Knowledge hypergraphs surpass traditional binary knowledge graphs by encapsulating complex $n$-ary atomic facts, providing a more comprehensive paradigm for semantic representation. However, constructing high-quality hypergraphs remains challenging due to the \textit{scenario gap}: generic extractors struggle to generalize across diverse domains with specific jargon, while existing methods often fail to balance structural skeletons with fine-grained details. To bridge this gap, we propose \textbf{Hyper-KGGen}, a skill-driven framework that reformulates extraction as a dynamic skill-evolving process. First, Hyper-KGGen employs a \textit{coarse-to-fine} mechanism to systematically decompose documents, ensuring full-dimensional coverage from binary links to complex hyperedges. Crucially, it incorporates an \textit{adaptive skill acquisition} module that actively distills domain expertise into a Global Skill Library. This is achieved via a stability-based feedback loop, where extraction stability serves as a relative reward signal to induce high-quality skills from unstable traces and missed predictions. Additionally, we present \textbf{HyperDocRED}, a rigorously annotated benchmark for document-level knowledge hypergraph extraction. Experiments demonstrate that Hyper-KGGen significantly outperforms strong baselines, validating that evolved skills provide substantially richer guidance than static few-shot examples in multi-scenario settings.