If for every operator represented by $A$ w.r.t to a basis $\mathcal{B}$, the matrix representation of $T^*$ is $A^*$, then $\mathcal{B}$ is orthogonal

Question

Let $V$ be a finite-dimensional inner product space. Assume that for every linear operator $T$, represented by $A$ w.r.t to a basis $\mathcal{B}$, the matrix representation of the adjoint w.r.t to $\mathcal{B}$ is $A^*$. Prove $\mathcal{B}$ is an orthogonal basis of vectors with the same norm (not necessarily 1).

I'm really stuck on this one, I have no lead.

The main thing I know is that I need to show that for every $b_i,b_j\in\mathcal{B},\ i\neq j \Rightarrow \langle b_i,b_j\rangle = 0$, and $\langle b_i,b_i\rangle = a,\ a\in\mathbb{R}$.

Any hints on this one?