Because a tensor is a multilinear form, it can be fully specified by how it act on all combinations of basis sets, which can be done in terms of components. We refer to each component as:
where we remember that the raised indices refer dual vector.

$T_{i_{1}…i_{m}}=T(e_{i_{1}},…,e_{i_{m}},e_{j_{1}},…,e_{j_{m}})$

Some examples: