Most existing benchmarks evaluate social bias in large language models (LLMs) using multiple-choice questions (MCQ) grounded in stereotypical contexts. While MCQ formats enable straightforward quantification, how LLMs manifest social bias in task-specific contexts and under different output constraints remains underexplored. We introduce SCENE, a benchmark for measuring the social bias shown by LLMs in scenario-completion tasks. Given short scenarios involving social groups, models must complete key elements of the situation through two tasks: (i) Character Specification, which asks models to assign social attributes to characters, and (ii) Plot Continuation, which asks models to select the most plausible narrative continuation. Combining templates adapted from prior benchmarks with newly collected templates through a university-wide competition, we construct a dataset of over 75,000 Korean instances. Experiments show that bias measurements are highly sensitive to task context and output constraints, even for identical underlying scenarios, suggesting that widely used MCQ-based evaluations may not fully capture certain manifestations of stereotypes. SCENE is publicly available at https://github.com/DSAIL-SKKU/SCENE-EMNLP-2026.