Abstract: It has been known since the 19th century that a smooth surface of degree 2 (resp. 3) in complex projective 3-space can be described as the zero locus of a 2 by 2 (resp. 3 by 3) determinant of linear forms. The somewhat more recent Noether-Lefschetz theorem implies that this does not extend to the general smooth surface of degree 4 or higher. One can ask for the ``next best thing" as follows: given a smooth surface X of degree d at least 4, does there exist a 2d by 2d skew-symmetric matrix M of linear forms such that X is the zero locus of the square root of det(M)? In this talk, I will discuss a result which gives an affirmative answer for smooth surfaces of degree 4, and how it can be applied to construct finite-dimensional irreducible representations of the generalized Clifford algebra associated to a ternary quartic form f=f(x,y,z) (i.e. the freest associative algebra over which f is the fourth power of a linear form). This is joint work with Emre Coskun (Tata Institute) and Rajesh Kulkarni (Michigan State University).