You can select a basis of unitary matrices with respect to which you can decompose your matrix. For example, if your matrix $A$ is $2^n\times 2^n$, then you can select the Pauli basis
$$
\sigma_y,\qquad y\in\{0,1,2,3\}^n
$$
You can find the decomposition very easily. Notice that if
$$
A=\sum_yA_y\sigma_y
$$
then calculating
$$
\text{Tr}(A\sigma_x)=A_x2^n
$$
because $A_x^2=I$ and the traces of all tensor products of Paulis except the all-identities tensor are 0.
Take, as an example, a matrix
$$
A=\left(\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1
\end{array}\right).
$$
Here, we have $n=2$, so we take the set of Pauli matrices
$$
\Lambda=\{I\otimes I,I\otimes X,I\otimes Y,I\otimes Z,X\otimes I,X\otimes X,X\otimes Y,X\otimes Z,Y\otimes I,Y\otimes Y,Y\otimes Z,Z\otimes I,Z\otimes X,Z\otimes Y,Z\otimes Z\}.
$$
For each in turn ($\sigma\in\Lambda$) you just calcuate $\text{Tr}(\sigma A)/4$. For instance,
$$
\text{Tr}(X\otimes X\cdot A)=2
$$
so $A_{1,1}=1/2$. Ultimately, you find out that
$$
A=\frac{1}{2}(X\otimes X-Y\otimes Y)+\frac14Z\otimes Z+\frac34I\otimes I+\frac{1}{4}(Z\otimes I-I\otimes Z)
$$
Of course, if you want to ask a question along the lines of the smallest set of unitaries with which you can decompose a specific $A$, that might be a very different question!