Orthogonal representation

Definition
Let $$G$$ be a group and $$k$$ a field. An orthogonal representation of $$G$$ over $$k$$ is a homomorphism $$\rho:G \to O(n,k)$$ where $$O(n,k)$$ is the orthogonal group of order $$n$$ over the field k (for some finite $$n$$).

Since $$O(n,k)$$ is a subgroup of $$GL(n,k)$$, every orthogonal representation can be viewed as a linear representation.