L(x) = b x X = [x3 0 -x1]T (Using definition of Cross Prod. from wikipedia ;) See comment below for details )
L(x) = M(3x3) x [x1 x2 x3]T
Now, what should M do with X? :D :D
Exchange the top and bottom rows, place sign, nullify middle row.
That can be achieved with M:
[0 0 1
0 0 0
-1 0 0]
Its Eigenvalues turn out to be: 0, +i. -i (A)