In "Algebraic geometry: a first course", by Harris, Grassmannian is described, under the Plucker embedding, as the locus of totally decomposable vectors in the projectivization of the exterior power $\bigwedge^k V$. Here
Decomposable elements of $\Lambda^k(V)$
a characterization of $m$-decomposable vectors in $\bigwedge^k V$ is given; however, I am struggling to find any reference for the locus of $m$-decomposable vectors in $\mathbb{P}(\bigwedge^k V)$. I fell like it should be a variety, not necessarily smooth, that contains the Grassmannian $G(k,V)$ for every choice of $m$. I am particularly interested in the case of $1$-decomposable vectors in $\mathbb{P}(\bigwedge^3 \mathbb{C}^5)$, but I think there should exist a fancy treatment of the topic. Can you provide me some reference?
If a 2-vector is 1-decomposable, it is also 2-decomposable (more generally, if a $k$-vector is $(k-1)$-decomposable, it is $k$-decomposable). In particular, the locus of 1-decomposable vectors in $\mathbb{P}(\wedge^2\mathbb{C}^5)$ is $G(2,5)$.
EDIT. On the other hand, the action of the group $\mathrm{PGL}(\mathbb{C}^5)$ on $\mathbb{P}(\wedge^2\mathbb{C}^5)$ has only two orbits, the Grassmannian and its complement, and since the locus of 1-decomposable vectors is obviously $\mathrm{PGL}(\mathbb{C}^5)$-invariant, it follows that it is equal to the entire space $\mathbb{P}(\wedge^2\mathbb{C}^5)$.