Let $G(m,n)$ be a real oriented Grassmanian of oriented $m$ planes in $\mathbb{R}^{n+m}$. Real de Rham cohomology classes of this space can be represented by $SO(n+m)-$invariant differential forms. In which cases these forms are algebraic in Plucker coordinates?

Example (m=1) The invariant volume form on $G(1,n-1)=\mathbb{S}^{n-1}$ is $$\dfrac{p_1dp_2 \wedge dp_3 \wedge ... \wedge dp_n-p_2dp_1 \wedge dp_3 \wedge ... \wedge dp_n+...+(-1)^{n-1}p_n dp_1 \wedge dp_2 \wedge ... \wedge dp_{n-1} }{(\sum{p_i^2})^{n/2}}.$$

Example (m=2) Invariant volume form on $Gr(2,4)$ has such a presentation, as was pointed out in comments to my previous question, see here. The denominator is equal to $(\sum{p_{ij}^2})^{3}.$


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