Let $V,W$ be two finite dimensional Hilbert spaces over $\mathbb{R}$. Suppose that $\{v_1,\cdots,v_m\}$ and $\{w_1,\cdots,w_n\}$ are orthonormal subsets of $V$ and $W$, respectively. Define $T:V\to W$ by $T(v):=\Sigma _{i=1}^ma_i\langle v,v_i\rangle w_i$ with $a_1,\cdots,a_m\in \mathbb{R}$. My question is: are there basis of $V$ and $W$ such that the matrix associated with $T$ with respect to these basis can be represented in a nice way?
For instance, suppose that $V=W=\mathbb{R}^p$, $v_i=w_i$ and $a_i=1$ for all $i\in \{1,\cdots,m\}$. Then $T$ is the orthonormal projection of $\text{Span}(v_1,\cdots,v_m)$ which implies that the matrix associated with $T$ with respect to the canonical basis of $\mathbb{R}^p$ is $A(A^TA)^{-1}A$ with $A$ being the matrix whose columns are $v_1,\cdots,v_m$.
I tried to extend those orthonormal subsets to basis in order to accomplish what I want, but I failed.
Any help is highly appreciated!