My question is: is there anywhere in tensors that we lose something by dropping the basis, or where it makes something more difficult? Like by saying the a tensor $T^{ij}e_i\otimes e_j$ is represented entirely by the notation $T^{ij}$?
Another thing, because there is only one i and j in the previous expression, does the summation convention kick in?
I heard this on the YouTube channel XylyXylyX (that's actually the name). Correct me if this is wrong because I've just started learning this myself.
Strictly speaking $\mathbf T^{ij}$ represents the $(i,j)$th component of the $(2,0)$ tensor $\mathbf T$ relative to a particular basis of $V \times V$, and the whole tensor would be represented by
$\mathbf T^{ij} \mathbf e_i \otimes \mathbf e_j$
which is short for
$\sum_{i=1}^n\sum_{j=1}^n\mathbf T^{ij} \mathbf e_i \otimes \mathbf e_j$
But $\mathbf T^{ij}$ is often used informally to represent all of the $n^2$ components of $\mathbf T$.