Conversations often adhere to well-understood social norms that vary across cultures. For example, while "addressing parents by name" is commonplace in the West, it is rare in most Asian cultures. Adherence or violation of such norms often dictates the tenor of conversations. Humans are able to navigate social situations requiring cultural awareness quite adeptly. However, it is a hard task for NLP models. In this paper, we tackle this problem by introducing a "Cultural Context Schema" for conversations. It comprises (1) conversational information such as emotions, dialogue acts, etc., and (2) cultural information such as social norms, violations, etc. We generate ~110k social norm and violation descriptions for ~23k conversations from Chinese culture using LLMs. We refine them using automated verification strategies which are evaluated against culturally aware human judgements. We organize these descriptions into meaningful structures we call "Norm Concepts", using an interactive human-in-loop framework. We ground the norm concepts and the descriptions in conversations using symbolic annotation. Finally, we use the obtained dataset for downstream tasks such as emotion, sentiment, and dialogue act detection. We show that it significantly improves the empirical performance.