If the dimension of a vector space is odd, then all (orientation-preserving) rotations in odd dimensions fix some axis.  Many of the differences between even-dimensional and odd-dimensional geometry relate to this fact.  For example,

 - The lack of symplectic structure in odd dimensions follows from the Lie-algebra version of the above statement: all odd-dimensional antisymmetric maps are degenerate.
 - The $-1$ map doesn't fix any axis, so it cannot be orientation-preserving in odd dimensions.
 - [Synge's theorem][1] states that if $M$ is compact, Riemannian, and has positive sectional curvature, then there is a conclusion which depends on the pairity of its dimension.  The proof makes essential use of the above fact. (See Lemma 3.8 in "Riemannian Geometry" by do Carmo.)


  [1]: https://en.wikipedia.org/wiki/Synge%27s_theorem