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):

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 we ask:

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