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So this is how tic tac toe would work if it was in n-space. I think. Basically you must claim n+1 continuous spaces in some cross section or throughout the entire n-space. Being continuous means that the closest next space is no more than 1 location away in each coordinate but is also at least 1 location away in at least 1 coordinate (otherwise it'd be the same point).
Example winning paths in 4 space:
2d - CS 3d - CS No CS 1d - CS
(0.1.2.3) (2.2.2.2) (0.0.0.0) (0.1.2.3)
(0.2.2.2) (1.2.1.1) (1.1.1.1) (1.1.2.3)
(0.3.2.1) (0.2.0.0) (2.2.2.2) (2.1.2.3)