For matrices larger than 3X3, you can simply extend the process which you used for 3X3 matrices. It shouldn't be too difficult (although it can be very tedious...).
Just as for 3X3 matrices you reduced to a set of three 2X2 matrices (with coefficients), you can do the same for an nXn matrix (reducing to set of n-1Xn-1 matrices, each of which you reduce to a set of n-2Xn-2 matrices and so on untll you arrive at (many!) 2X2 matrices which you can evaluate directly). In fact, the definition of determinant is often giving inductively.