How do I expand this? I am self studying Mathematical Methods for Physicists by Arfken. I came across an equation involving the summation of the Levi Civita symbol:$\sum_k \epsilon_{ijk} ê_k$

I'm not sure how to deal with such sums. Could someone please expand this for me so I can understand how this really works?
Thanks!
 A: I think the full expression from Arfken you're referencing is about the cross product of Cartesian basis vectors:
$$\hat{\mathbf{e}}_i\times\hat{\mathbf{e}}_j=\sum_{k}\varepsilon_{ijk}\hat{\mathbf{e}}_k$$
This notation is a general expression that works for all $i,j,k\in\{1,2,3\}$. Let's do a few examples:
$$\hat{\mathbf{e}}_1\times\hat{\mathbf{e}}_1=\varepsilon_{111}\hat{\mathbf{e}}_1+\varepsilon_{112}\hat{\mathbf{e}}_2+\varepsilon_{113}\hat{\mathbf{e}}_3=\mathbf{0}$$
$$\hat{\mathbf{e}}_1\times\hat{\mathbf{e}}_2=\varepsilon_{121}\hat{\mathbf{e}}_1+\varepsilon_{122}\hat{\mathbf{e}}_2+\varepsilon_{123}\hat{\mathbf{e}}_3=\hat{\mathbf{e}}_3$$
$$\hat{\mathbf{e}}_1\times\hat{\mathbf{e}}_3=\varepsilon_{131}\hat{\mathbf{e}}_1+\varepsilon_{132}\hat{\mathbf{e}}_2+\varepsilon_{133}\hat{\mathbf{e}}_3=-\hat{\mathbf{e}}_2$$
In other words, what this notation tells us is that $\hat{\mathbf{e}}_i\times\hat{\mathbf{e}}_j=\hat{\mathbf{e}}_k$ whenever $\{i,j,k\}$ is an even permutation of $\{1,2,3\}$, $\hat{\mathbf{e}}_i\times\hat{\mathbf{e}}_k=-\hat{\mathbf{e}}_j$ whenever $\{i,k,j\}$ is an odd permutation of $\{1,2,3\}$, and $\hat{\mathbf{e}}_i\times\hat{\mathbf{e}}_i=\mathbf{0}$.
