I learned from the following post that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is diffeomorphic to $\mathbb{R}P^{n+1}\setminus \{pt\}$.(There is  a natural embedding$([x],v)\mapsto [x,x.v]$):

https://math.stackexchange.com/questions/1486105/is-the-total-space-of-the-tautological-line-bundle-over-mathbbrpn-a-non


Now assume that $E$ is the total space of the tautological 2-plane bundle over the real Grassmanian $G(2,n)$.


As  a generalization of the above fact we ask:

>Is there  a  (natural) embedding of $E$ into $G(2,n+1)$? If yes,  what is the topological type of the remainder $G(2,n+1)\setminus E$. Is it possible to have an embedding with a one point remainder?