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A single math equation does not have a determinant. A system of equations (3x3 , 4x4, etc.) will have a determinant. You can find a determinant of a system by converting the system into a corresponding matrix and finding its determinant.
The determinant function is only defined for an nxn (i.e. square) matrix. So by definition of the determinant it would not exist for a 2x3 matrix.
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In theory, a 2x2 determinant requires the evaluation of 2 products, a 3x3 determinant requires 6 products, a 4x4 determinant requires 24 products (note: that is the factorial function). The Rule of Sarrus is just a convenient memory aid for this specific case.