Despite recent progress, large language models (LLMs) still face the challenge of appropriately reacting to the intricacies of social and cultural conventions. This paper presents Mango, a methodology for distilling high-accuracy, high-recall assertions of cultural knowledge. We judiciously and iteratively prompt LLMs for this purpose from two entry points, concepts and cultures. Outputs are consolidated via clustering and generative summarization. Running the Mango method with GPT-3.5 as underlying LLM yields 167K high-accuracy assertions for 30K concepts and 11K cultures, surpassing prior resources by a large margin in quality and size. In an extrinsic evaluation for intercultural dialogues, we explore augmenting dialogue systems with cultural knowledge assertions. Notably, despite LLMs inherently possessing cultural knowledge, we find that adding knowledge from Mango improves the overall quality, specificity, and cultural sensitivity of dialogue responses, as judged by human annotators. Data and code are available for download.