A difference set D in a group G is called a skew Hadamard difference set (in short SHDS) if G is the disjoint union of D, the set of inverses of D, and the identity element. We construct a new family of skew Hadamard difference sets in the additive group of F_{3^{2h+1}}, by using a class of permutation polynomials of F_{3^{2h+1}} obtained from the Ree-Tits slice symplectic spreads in the three-dimensional projective space over F_{ 3^{2h+1}}. Our new construction and the Ding-Yuan construction provide the only known ``non-classical'' examples of SHDS.

Also we generalize the recent Ding-Yuan construction of SHDS by using planar functions. This generalization also gives rise to new constructions of strongly regular graphs.