As a bundle it fibers over $\mathbb RP^2$ with fiber $\mathbb R^3 \setminus \mathbb R^2$.
I think the most natural way to describe this space would be to consider two bundles over $\mathbb RP^2$: (1) $\mathbb RP^2 \times \mathbb R^3$ and (2) $\{(L,v) : L \in \mathbb RP^2, v \in \mathbb R^3, v \perp L \}$ for this purpose $\mathbb RP^2$ is considered the space of lines in $\mathbb R^3$.
So the total space of bundle (2) is a subspace of the total space of bundle (1). The bundle you're interested in, is the complement of (2) in (1).
So another way to say what this space is, is it's all pairs $(L, v)$ such that $L$ is in $\mathbb RP^2$ and $v$ projects to a non-zero vector in $L$ (using orthogonal projection).
So another way to describe it would be the tangent bundle of $\mathbb RP^2$ fiber-product with a certain bundle over $\mathbb RP^2$ -- as a bundle over $\mathbb RP^2$ this "certain bundle" is the map $\mathbb R \times S^2 \to \mathbb RP^2$ where $(p,q) \longmapsto \pi(q)$ where $\pi : S^2 \to \mathbb RP^2$ is the 2:1 covering map.
Even simpler, as a space it's diffeomorphic to $S^2 \times \mathbb R^3$. You can make this equivarient with respect to the left $GL(3)$ action as well -- $GL(3)$ has its standard (projectivized) action on $S^2$, similarly it acts on $\mathbb R^3$.