I have a book that says

Define the 9 Cartesian components of a second order Tensor T by $\displaystyle T_{ij}=e_i.T.e_j$.

In component form, this can be written as $\displaystyle T=T_{ij}e_ie_j$.

This can be generalised for higher order tensors. For example, if B is a tensor of order 3 then $\displaystyle B=B_{ijk}e_ie_je_k$.
My first question is, how are they defining their cartesian components for a third order tensor?

Is there a general rule for defining cartesian components in a k-th order tensor?

Thankyou to anyone who posts!