It's well known that the even-dimensional complex quadric $Q_{2n}$, defined by the equation $z_1^2+\cdots +z_{2n+2}^2=0$ in complex projective space $CP^{2n+1}$, is diffeomorphic with the Grassmann manifold of oriented real 2-planes. A natural diifeomorphism is given as follows: for any two orthogonal vectors $v_1$ and $v_2$ in $R^{2n+2}$, the oriented 2-plane spanned by $v_1, v_2$ corresponds to $\pi(v_1+iv_2)$ where $\pi$ is the natural projection $C^{2n+2}\backslash \{0\}\rightarrow CP^{2n+1}$.

My question is: what part of this breaks down for the odd-dimensional quadrics?