Does $SU(N)$ have a pseudo-real representation? If so, how can we construct them explicitly? I am looking specifically for pseudo-real representation instead of real representation (The adjoint representation of $SU(N)$ is real).

If $g\in SU(N)$ and $R(g)$ is a pseudo-real representation, then there exists a matrix $C$ such that

$$\bar{R}(g)=CR(g)C^{-1},$$

where $\bar{R}(g)$ means complex conjugation and $C$ is not the identity matrix.

I would appreciate any comment or reference. Thank you!