  We prove the formal degree conjecture for simple supercuspidal
representations of symplectic groups and quasi-split even special orthogonal
groups over a p-adic field, under the assumption that p is odd. The essential
part is to compute the Swan conductor of the exterior square of an irreducible
local Galois representation with Swan conductor 1. It is carried out by passing
to the equal characteristic local field and using the theory of Kloosterman
sheaves.
