We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n⁴) where n is the size of the matrix. We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold.