Building embodied autonomous agents capable of participating in social interactions with humans is one of the main challenges in AI. Within the Deep Reinforcement Learning (DRL) field, this objective motivated multiple works on embodied language use. However, 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. 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. As a first step, we propose to expand current research to a broader set of core social skills. To do this, we present SocialAI, a benchmark to assess the acquisition of social skills of DRL agents using multiple grid-world environments featuring other (scripted) social agents. We then study the limits of a recent SOTA DRL approach when tested on SocialAI and discuss important next steps towards proficient social agents. Videos and code are available at https://sites.google.com/view/socialai.