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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)
(0.4.2.0) (4.2.4.4) (3.3.3.3) (3.1.2.3)
(0.0.2.4) (3.2.3.3) (4.4.4.4) (4.1.2.3)
1d - CS: (0.1.2.3), (1.1.2.3), (2.1.2.3), (3.1.2.3), (4.1.2.3)
2d - CS: (0.1.2.3), (0.2.2.2), (0.3.2.1), (0.4.2.0), (0.0.2.4)
3d - CS: (2.2.2.2), (1.2.1.1), (0.2.0.0), (4.2.4.4), (3.2.3.3)
No CS: (0.0.0.0), (1.1.1.1), (2.2.2.2), (3.3.3.3), (4.4.4.4)
CS = Cross Section, so a 3d cross section means that in larger space than 3 space only three dimensions can change. It could be 1,2,4 in 5 space or 2,3,7 in 10 space, it doesn't matter which three just that is three.