Let $O(n)$ and $SO(n)$ denote the split orthogonal linear algebraic group and its special subgroup, over some fixed field of characteristic not two. 
I am looking for a reference that explains how to describe the simple (finite-dimensional) representations of $O(n)$ in terms of the simple representations of $SO(n)$.

The relation between the complex representations of the corresponding compact Lie groups is explained in section VI.7 of  Bröcker and tom Dieck’s *Representations of Compact Lie Groups*.  I believe the key statements made there also hold in the above algebraic situation,  and I have worked this out in a fair amount of detail,  but I am currently unwilling to believe that this has not already been done somewhere in the literature.