I am trying the following proof. If objects X and Y transform as vectors under orthogonal transformation, then the cross product of X and Y also transforms as vector. So I try this way;
Let
$$x_{j}^{'} = a_{jn}x_{n}$$ and $$y_{k}^{'} = a_{km}y_{m}$$
Assume: $$\vec{z} = \vec{x} \times \vec{y}$$
I would like to show the ith component of the z is as follow (hence complete the proof);
ie. $$z_{i}^{'} = (\vec{x}\times \vec{y})_{i} = a_{ij} z_{j}$$
I started like below (I understand here): $$z_{i}^{'} = \varepsilon_{ijk} x_{j}^{'}y_{k}^{'} = \varepsilon_{ijk}a_{jn}x_{n}a_{km}y_{m} $$
Then according to a book, it says; $$ = \varepsilon_{ijk}a_{jn}x_{n}a_{km}y_{m} = \varepsilon_{nml}a_{il}x_{n}y_{m} = a_{ij}\varepsilon_{jnm}x_{n}y_{m} = a_{ij}z_{j}$$
I am bit confused how these levi civita tensor is changing to nml, $$\varepsilon_{nml}$$ and the next step.
Thanks for any input.