$K$-theory of real Grassmann manifolds

Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb{R}^n$. We compute the complex $K$-ring of $G_{n,k}\:$, up to a small indeterminacy, for all values of $n,k$ where $2 \leqslant k \leqslant n - 2$. When $n \equiv 0 (\operatorname{mod} 4), k \equiv 1 (\operatorname{mod} 2)$, we use the Hodgkin spectral sequence to determine the $K$-ring completely.