I have some components of a contravariant rank two tensor with respect to the standard basis $\{ \boldsymbol{e_1}\ \boldsymbol{e_2} \}$ which are called $a^{ij}$. The task is to find the components $a_{ij}$ with respect to the dual basis of the standard basis. The textbook solution to this one is: $$\boldsymbol{A} = a^{ij}|\boldsymbol{e_i}\rangle\langle\boldsymbol{e_j}|=a^{ij}g_{ik}g_{lj}|\boldsymbol{e^k}\rangle\langle\boldsymbol{e^{l}}|= a_{kl}|\boldsymbol{e^k}\rangle\langle\boldsymbol{e^{l}}|$$ from which follow that $a_{kl} = a^{kl}$.
I get the part of using the metric tensors to lower the indices and the fact that $g_{ij} = \delta_{ij}$, but what I really struggle with is the first part of the calculation where it says: $$\boldsymbol{A} = a^{ij}|\boldsymbol{e_i}\rangle\langle\boldsymbol{e_j}|$$ I honestly never saw this in my scriptum and have some troubles understanding the meaning of it. Does this equation just represent the fact that $\boldsymbol{A}$ is a linear combination of the components $a^{ij}$ and the basis vectors $\boldsymbol{e_1}$ and $\boldsymbol{e_2}$?.
Would be really cool if someone could help me out with this!