The determinante of $\left(A_{ij}\right)$ is defined by Leibniz as
$$\det \left(A_{ij}\right) := \sum_{\sigma}\text{sign}(\sigma)\Pi_i A_{i,\sigma(i)} $$
With permutations $\sigma$ and their sign $\text{sign}(\sigma)$ ($=+1$ if Permutation is an even of $1,2,3,4,...$, $-1$ if pertutation is odd, $0$ else)
This can be simplified with
$$\det \left(A_{ij}\right)=\epsilon_{i_i,i_2,..}A_{1,i_1}A_{2,i_2}A_{3,i_3}\cdots $$
Introducing base vectors $e_i$ of a $n$ dimensional vector space with $i=1,2,...,n$ and $e_ie_j=\delta_{ij}$ (Here $n=3$)
Than you can write your Matrix
$$\begin{pmatrix} \delta_{il} & \delta_{im} & \delta_{in} \\
\delta_{jl} & \delta_{jm} & \delta_{jn} \\
\delta_{kl} & \delta_{km} & \delta_{kn}\end{pmatrix}=
\begin{pmatrix}e_i^\top\\e_j^\top\\e_k^\top\end{pmatrix}\begin{pmatrix}e_l&e_m&e_n\end{pmatrix}$$
The determinant of a matricies build of vectors is now
$$\det\begin{pmatrix}e_i^\top\\e_j^\top\\e_k^\top\end{pmatrix}=\epsilon_{ijk}\qquad \det\begin{pmatrix}e_l&e_m&e_n\end{pmatrix}=\epsilon_{lmn}$$
Using the product rule of determinats
$$\det\begin{pmatrix} \delta_{il} & \delta_{im} & \delta_{in} \\
\delta_{jl} & \delta_{jm} & \delta_{jn} \\
\delta_{kl} & \delta_{km} & \delta_{kn}\end{pmatrix}=\det\begin{pmatrix}e_i^\top\\e_j^\top\\e_k^\top\end{pmatrix}
\cdot\det\begin{pmatrix}e_l&e_m&e_n\end{pmatrix}=
\epsilon_{ijk}\epsilon_{lmn} $$
PS. I am using the notation $\left(A_{ij}\right)$ is the matrix with bound indicies and $A_{ij}$ are their elements.
PPS. This calculation can also be found on http://de.wikipedia.org/wiki/Levi-Civita-Symbol
