Building embodied autonomous agents capable of participating in social interactions with humans is one of the main challenges in AI. This problem motivated many research directions on embodied language use. Current approaches focus on language as a communication tool in very simplified and non diverse social situations: the "naturalness" of language is reduced to the concept of high vocabulary size and variability. In this paper, we argue that aiming towards human-level AI requires a broader set of key social skills: 1) language use in complex and variable social contexts; 2) beyond language, complex embodied communication in multimodal settings within constantly evolving social worlds. In this work we explain how concepts from cognitive sciences could help AI to draw a roadmap towards human-like intelligence, with a focus on its social dimensions. We then study the limits of a recent SOTA Deep RL approach when tested on a first grid-world environment from the upcoming SocialAI, a benchmark to assess the social skills of Deep RL agents. Videos and code are available at https://sites.google.com/view/socialai01 .