If we have a 2 x 2 matrix
the determinant is given by
If we have a 3 x 3 matrix
the determinant is given by
Is there such a formula for a 4 x 4 matrix
The procedure you used to find the determinant of the 3x3 is called Sarrus' Rule. This cannot be applied to find out the determinant of a 4x4 matrix.
You can use the method of cofactor expansion to find the determinant of a 4x4.
This link gives you the formula for the 4x4 determinant