Suppose $X$ is the orthogonal Grassmanian. We know the Plücker embedding does not span the whole background $\mathbb{CP}^n,$ just a subspace $\mathbb{CP}^m.$

My question: is there an expression of the isometric embedding of $X$ into $\mathbb{CP}^m$?

By expression, I mean a way to write down explicitly the coordinates such as Plücker coordinates for the Grassmannian?