Is there relation between Grassmann Manifold and Grassmann Algebra?

Question

I'm an EE student and I'm beginning to learn about the Grassmann Manifold.
As is known that the Grassmann Manifold is a space treating each linear subspace with a specific dimension in the vector space $V$ as a single point, for example we can represent the set of all $k$-dimensional linear subspaces $X$ in the n-dimensional vector space $V$ as Grassmann Manifold $Gr(k,n)$, and treat each $X\in Gr(k,n)$ as a point in this local Euclidean space.
When I'm exploring materials about Grassmann Manifold, lots of discussions about the so-called Grassmann Algebra just come to me. I learned that Grassmann Algebra deals with a so-called Exterior Product and is difined on a space that is different from the common vector spaces.
So I'm wondering about the relationship between Grassmann Manifold and Grassmann Algebra? Is there any algebra defined for the space of Grassmann Manifold?