In the linear algebra course I took vector spaces where introduced with a (commutative) field. The classical theorems are proven under this assumption.
However, I was wondering what implications it would have if one would only consider skew fields. I have looked it up and the commutativity is used in several proofs such as "every set of $n+1$ vectors is linearly dependent in a $n$-dimensional vector space".
Do all the classical results still hold? Or are their major modifications?
(I'm asking this question in the context of affine geometry, where every affine space of dimension $\geq 3$ can be considered as a vector space with a skew field)