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$([x],v)\mapsto [x,x.v]$):

http://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?