We study the injective norm of random skew-symmetric tensors and the associated fermionic quantum states, a natural measure of multipartite entanglement for systems of indistinguishable particles. Extending recent advances on random quantum states, we analyze both real and complex skew-symmetric Gaussian ensembles in two asymptotic regimes: fixed particle number with increasing one-particle Hilbert space dimension, and joint scaling with fixed filling fraction. Using the Kac--Rice formula on the Grassmann manifold, we derive high-probability upper bounds on the injective norm and establish sharp asymptotics in both regimes. Interestingly, a duality relation under particle--hole transformation is uncovered, revealing a symmetry of the injective norm under the action of the Hodge star operator. We complement our analytical results with numerical simulations for low fermion numbers, which match the predicted bounds.