The monodromy groups of the n-th order hypergeometric differential equations are characterized as the subgroups of the general linear group GL(n), which are generated by the companion matrices of two monic co-prime polynomials f (x) and g(x) of degree n. If we impose some conditions on the polynomials f (x), g(x) (like f (x) and g(x) are not simultaneously polynomials in higher powers of x, and they are self-reciprocal), then the associated monodromy group preserves either a non-degenerate symplectic form (this happens only when n is even, and f (0) = g(0) = 1) or a non- degenerate quadratic form (and this happens when f (0) = 1, g(0) = −1), and it is contained in the respective symplectic group (of the corresponding symplectic form) or in the orthogonal group (of the corresponding quadratic form) as a Zariski dense subgroup.