The phrase "keeping under one hat" suggests keeping things secret or within a small group. Some antonyms for this phrase could include sharing, revealing or broadcasting. These words imply a sense of openness, transparency, and distribution of information. Other antonyms could include decentralizing, delegating or dispersing, where information or power is spread out among a larger group of people. In contrast, the idea of keeping things under one hat suggests a concentration of power or information, which may not always be the most effective or honest approach. Antonyms for this phrase can encourage collaboration, communication, and a culture of sharing among individuals or teams.