Consider a linear map $\phi\colon V \rightarrow W$ and bases $v_1, \dots, v_n$ and $w_1, \dots, w_m$ of $V$ and $W$, respectively.
How do we define the matrix of the linear map $A$ such that $\phi(x) = Ax,$ using those vectors? I've searched online and I am having trouble locating a definition.
It's not quite correct to write $\phi(x) = Ax$. Whatever $x$ is, it's an element of an abstract vector space, and it doesn't make sense to multiply $x$ by a matrix $A$. Instead, you should write $$ [\phi(x)]_{\beta} = A [x]_{\alpha}, $$ where:
Here is one way to remember what the matrix representation of $\phi$ is. What is the minimal information you need to write down in order to fully describe $\phi$? If you tell me the values $\phi(v_1),\ldots,\phi(v_n)$, then you have fully described $\phi$. Given that information, I can now figure out the value of $\phi(x)$ for any vector $x \in V$ whatsoever.
But how do you write down $\phi(v_i)$? Since we're working in an abstract vector space, $\phi(v_i)$ might be some mathematical object which is not easy to write down. Instead, you should just write down the coordinate vector of $\phi(v_i)$ with respect to the basis $\beta$. So you write down the following information: $$ [\phi(v_1)]_{\beta},\, [\phi(v_2)]_{\beta},\,\ldots,\, [\phi(v_n)]_{\beta}. $$ You have now described $\phi$ fully. What you just wrote down is the matrix representation of $\phi$: $$ A = \begin{bmatrix} [\phi(v_1)]_{\beta} & [\phi(v_2)]_{\beta} &\ldots & [\phi(v_n)]_{\beta} \end{bmatrix}. $$