Large language models perform well on many logical reasoning benchmarks, but it remains unclear which core logical skills they truly master. To address this, we introduce LogicSkills, a benchmark that isolates three fundamental logical skills: (i) formal symbolization\unicode{x2013}translating premises into first-order logic; (ii) countermodel construction\unicode{x2013}showing that an argument is logically invalid by constructing a finite countermodel; and (iii) validity assessment\unicode{x2013}determining whether a conclusion follows from a set of premises. Items are drawn from the two-variable fragment of first-order logic without identity and are presented in both English and a Carrollian nonce-word language. All instances are solver-verified with Z3 for correctness and non-triviality. Across conventional instruction-tuned LLMs, performance is high on validity assessment but substantially lower on formal symbolization and countermodel construction, highlighting that high task-level accuracy can mask weaknesses in core logical skills. In contrast, recent reasoning-tuned models perform strongly across all three tasks, suggesting a more systematic logical skill profile.