Please, don't do that! I've seen such notation from time to time even in textbooks, but this thread shows, how confusing that is! It's of course not wrong, but it's confusing. My experience is that it is much better to keep to bra-ket notation strictly for the abstract vectors. It's the great advantage of the Dirac notation that it is representation independent, and you can do many general calculations in a much clearer way without introducing a basis/representation.
If you have an application, where matrix-vector notation in ##\mathbb{C}^d## or ##\ell^2## are needed, it's very easy to introduce them. I suggest then you write
$$\psi_n=\langle n|\psi \rangle,$$
where the ##|n \rangle## are an orthonormal basis and then define
$$\psi=\begin{bmatrix}\psi_1 \\ \psi_2\\ \vdots \end{bmatrix},$$
and for the Operators you introduce matrix elements
$$A_{mn}=\langle m |\hat{A}|n \rangle$$
and matrices
$$\boldsymbol{A}=\begin{pmatrix} A_{11} & A_{12} & \ldots \\
A_{21} & A_{22} & \ldots & \\
\vdots & \vdots & \ddots & \end{pmatrix}.$$
This, perhaps somewhat overpedantic, notation helps a lot to avoid the kind of confusion we try to cure in this thread!