Developmental psychologists have long-established the importance 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 suite of procedurally generated environments, which simplifies conducting experiments regarding those concepts. We show examples of such experiments with RL agents and pure-text 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.