A relationship between Kronecker's Delta and Ricci's Tensor

[fireball]

I admit that most of the procedure is computational, but such
a relationship comes to aid a lot when proving propositions
of vector and tensor algebra arising from the
framework of solid and continuum mechanics.

Let \(\{\mathbf e_1, \mathbf e_2, \mathbf e_3\}\) be an orthonormal basis
for a three-dimensional Euclidean vector space \(\mathcal V\),
endowed with inner product \(\cdot\) and vector product \(\times\).
Assume Einstein's convention, according to which the summation symbol
is suppressed, and summation over all possible values of an index
is signaled implicitly by the fact that it occurs twice in a monomial term. Show that:

\(\displaystyle e_{ijk}e_{hmk} = \delta_{ih}\delta_{jm} - \delta_{im}\delta_{jh},\)

where

\(\displaystyle e_{ijk} := \mathbf e_i \times \mathbf e_j\cdot \mathbf e_k\quad (i,j,k=1,2,3),\)

and

\(\displaystyle \delta_{ij}:=\mathbf e_i \cdot \mathbf e_j\quad(i,j=1,2,3).\)