What's the minimal embedding of orthogonal Grassmannian? 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?
 A: The image under the Plücker embedding of the isotropic Grassmannian of $r$-planes in $\mathbb{C}^k$ (of course, $r\leq k/2$) spans the subspace generated under $SO(k)$ acting on the highest weight vector $v_1\wedge \cdots \wedge v_r$.  This is the entirety of $\bigwedge^{r}\mathbb{C}^k$ with the sole exception of when $k=2r$.  In that case, I believe it spans a half dimensional subspace, but to be honest, it is late, and I've forgotten the details.
A: I stumbled over here looking for something else. 
I think the short answer is to construct the orthogonal Grassmannian of isotropic n-planes in an 2n-dimensional space, take a list of all the principal pfaffians of a skew-symmetric n by n matrix. Since odd-pfaffians automatically vanish, the construction is slightly different in the even and odd cases. 
In general there is a construction of the minimal rational embedding of each compact hermitian symmetric space. In these cases the variety is the unique closed orbit of a highest weight vector and the minimal rational embedding is into the span of the orbit of the highest weight vector. For a complete picture see  Landsberg and Manivel's work https://arxiv.org/pdf/math/9902102.pdf 
