Interpretation of eigenvectors of cross product If we fix a non-zero vector $\boldsymbol{v}\in\mathbb{R}^3$, then the linear map $\boldsymbol{x}\mapsto\boldsymbol{v}\times\boldsymbol{x}$ has trivial eigenvectors $\boldsymbol{x}_1=t\boldsymbol{v}$ (where $t\neq 0$) with zero as its corresponding eigenvalue, and it is clear that there are no other real eigenvectors.
If one sets up the matrix for the mapping, and computes the other two eigenvectors, one finds that for any $t\neq 0$,
$$\boldsymbol{x}_2=t(v_1v_2-iv_3,-v_1^2-v_3^2,v_2v_3+iv_1)$$
and
$$\boldsymbol{x}_3=t(v_1v_2+iv_3,-v_1^2-v_3^2,v_2v_3-iv_1)$$
are eigenvectors corresponding to the eigenvalues $\pm i|\boldsymbol{v}|$.
My question is if there is some intuitive way to interpret the two complex eigenvectors. (For example, the complex eigenvectors of a $2\times 2$-rotation matrix have a nice geometric interpretation as the so called circular points in projective geometry. This is the kind of interpretation I seek.)

Update:
The linear map $\boldsymbol{x}\mapsto\boldsymbol{v}\times\boldsymbol{x}$ is the composition of a projection on the plane with normal $\boldsymbol{v}$, followed by a rotation by $\pi/2$ radians in that plane, and a correction of the length. The vectors $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$ above are clearly eigenvectors to the projection, and therefire they must also be eigenvectors of the rotation (since they are eigenvectors of the composition). This suggests a connection to the complex eigenvectors of a $3\times 3$-rotation matrix. I am still interested in interpretations of this.
 A: This is not quite an interpretation - rather a hint on possible generalizations.
One can look at vectors $\vec{v}=(v_x,v_y,v_z)$ in $\mathbb{R}^3$ as $2\times2$ traceless hermitian matrices $$\mathbf{v}=\left(\begin{array}{cc}v_z & v_x-iv_y \\ v_x+iv_y & -v_z\end{array}\right),$$ 
which form Lie algebra $su(2)$. Note that $\vec{u}\cdot \vec{v}=\frac12\mathrm{tr}\left(\mathbf{uv}\right)$ and $\vec{u}\wedge\vec{v}\mapsto\frac{1}{2i}[\mathbf{u},\mathbf{v}]$. 
Picking one vector $\vec{v}$ can be seen as fixing Cartan (maximal commutative) subalgebra of $su(2)$. Then two nontrivial eigenvectors will correspond to raising and lowering operators with respect to this subalgebra. In the structure theory of Lie algebras, such eigenvectors are called roots. Their study allows to give a complete classification of semisimple Lie algebras.
A: Here is a fully geometric view, showing in particular 
1) the deep analogy of operator $\left[n_{\times}\right]$ with its eigenvalues.
2) the fact that, using Cayley-Hamilton theorem, $$\left[n_{\times}\right]^3=-\left[n_{\times}\right]$$

