### Transforming a matrix into a system of equations

I have a $n \times n$ matrix that I would like to transform into a system of $n$ equations (for $n$ disctinct variables). I am not sure how to proceed (the end goal being to solve the system eventually). The number of variables is not fixed. Should I consider every row of the lattice and map each coefficient $c_{ij}$ to a string that I write as $"c_{11}x_1 + .....+ c_{1n-1}x_n + c_{1n}$. I am hoping there exists a more direct solution. Thanks!