We determine the rings of invariants $S^G$ where $S$ is the symmetric algebra on the dual of a vector space $V$ over ${\mathbb F}_2$ and $G$ is the orthogonal group preserving a non-singular quadratic form on $V$. The invariant ring is shown to have a presentation in which the difference between the number of generators and the number of relations is equal to the minimum possibility, namely $\dim V$, and it is shown to be a complete intersection. In particular, the rings of invariants computed here are all Gorenstein and hence Cohen-Macaulay.