The definition of zero-sum applies immediately to any number of players, as others have noted.
However, the nice properties of two-player zero-sum games, like the minmax theorem, unique value, interchangeability of equilibrium strategies, do not hold for 3 or more player zero-sum games unless extra conditions are imposed.
In particular note that one can take any two-player non-zero-sum (aka general-sum) game and add a third dummy player, whose actions have no effect on the original two players, and whose payoffs are set to make the game zero-sum. Now any equilibrium of this three-player game immediately gives an equilibrium of the original two-player game, so three-player zero-sum games are as hard to solve and at least as general as two-player general-sum games.
A class of many-player zero-sum games in which a number, but not all, of the nice properties persist is the case of zero-sum polymatrix games -- see:
Yang Cai, Ozan Candogan, Constantinos Daskalakis, Christos H. Papadimitriou:
Zero-Sum Polymatrix Games: A Generalization of Minmax. Math. Oper. Res. 41(2): 648-655 (2016).