You should at least understand that games are interactive, while formal systems are not. There are many ways around this problem, but don't be surprised if they make the result look less like the game you intended than a mathematical study of the game. You can make each permissible move a permissible instantiation of axiom schemata. You can batch all the moves of a particular game and make them axioms of a system. You can make a series of systems corresponding to all possible legal moves and games.
Theorems can describe individual moves, or entire board configurations, or entire games.
Many other variations are possible. Stern warnings before the core was turned in, and stern comments on the core, did not stop most students from turning in something that did not conform to the technical definition of a formal system. Rules of inference tell us what can be inferred from axioms and other theorems. They do not tell us what must be inferred. However, for metatheoretical purposes we can posit the set of all theorems, as if all permissible inferences were actually made.