For $\mathbf{Gr}(p,n)$ the $p$ dimensional subspace of $\mathbb{R}^n$, or equivalently $O(n)/O(p)\times O(n-p)$, a point has a unique projector representation $P = UU’$ where U is an $n \times p$ matrix whose columns form an orthonormal basis of the subspace. However, it would not be a unique representation for a point on the oriented Grassmannian $\mathbf{Gr}^+(p,n)=SO(n)/SO(p) \times SO(n-p)$, since the opposite orientations of the same subspace would result in the same $P$.
The question is then what could be a unique representation of $\mathbf{Gr}^+(p,n)$. I’ve read somewhere that one can simply use the tuple $(P, \pm1)$ to additionally indicate the orientation. However, I am concerned with the continuous time dynamics of a point on $\mathbf{Gr}^+(p,n)$. Assigning $\pm1$ would seem to introduce discontinuity, and it's unclear how to evolve this variable according to the dynamics.
Multivectors are one common tool for dealing with oriented subspaces.
Every oriented subspace $V\in\mathbf{Gr}^+(n,k)$ can be represented by a unique homogeneous unit multivector $v$, which can be written as $v=e_1\wedge\cdots\wedge e_k$ where $e_i,\cdots,e_k$ is any oriented orthonormal basis of $V$.