functional analysis - A linear map from $mathbb{C}^n otimes overline{mathbb{C}^n}$ to $M_n(mathbb{C})$

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}$

I am trying to establish some properties of the linear map: $T:\mathbb{C}^n \otimes \overline{\mathbb{C}^n} \to M_n(\mathbb{C})$ given by: $T(\ket{i} \otimes \ket{j}) = \ket{i}\bra{j} \text{ , } 0 \leq i,j \leq n-1$.

Where $\otimes$ denotes the tensor product, $\overline{\mathbb{C}^n}$ is the conjugate space of the complex (finite) Hilbert space $\mathbb{C}^n$, and $M_n(\mathbb{C})$ is the set of $n\times n$ matrices over $\mathbb{C}$.

I want to show the following:

(i) T is a bijection that preserves inner products (under the
Hilbert-Schmidt inner product on $M_n(\mathbb{C})$

(ii) $T(\ket{\psi}\otimes\ket{\varphi} = \ket{\psi}\bra{\varphi}$ for
every $\ket{\psi}, \ket{\varphi} \in \mathbb{C}^n$.

I think part of my trouble is figuring out the Dirac notation / how the outer product works. I'm not sure exactly what $\ket{i}\bra{j}$ looks like as a matrix (obviously it's a matrix in $M_n(\mathbb{C})$ but I'm not quite sure what the entries look like).

Also, I think part (ii) of the question is an immediate consequence from the fact that $T$ is a linear, inner product preserving bijection, but I'm a bit confused about why $\ket{\varphi}$ is now coming from $\mathbb{C}^n$ rather than $\overline{\mathbb{C}^n}$.

Any help is appreciated.