Developmental psychologists have long-established the importance of socio-cognitive abilities in human intelligence. These abilities enable us to enter, participate and benefit from human culture. AI research on social interactive agents mostly concerns the emergence of culture in a multi-agent setting (often without a strong grounding in developmental psychology). We argue that AI research should be informed by psychology and study socio-cognitive abilities enabling to enter a culture too. We discuss the theories of Michael Tomasello and Jerome Bruner to introduce some of their concepts to AI and outline key concepts and socio-cognitive abilities. We present The SocialAI school - a tool including a customizable parameterized uite of procedurally generated environments, which simplifies conducting experiments regarding those concepts. We show examples of such experiments with RL agents and Large Language Models. The main motivation of this work is to engage the AI community around the problem of social intelligence informed by developmental psychology, and to provide a tool to simplify first steps in this direction. Refer to the project website for code and additional information: https://sites.google.com/view/socialai-school.