Summary: Let \(K\) be a field admitting a cyclic Galois extension of degree \(n\). The main result of this paper is a decomposition theorem for the space of alternating bilinear forms defined on a vector space of odd dimension \(n\) over \(K\). We show that this space of forms is the direct sum of \((n-1)/2\) subspaces, each of dimension \(n\), and the non-zero elements in each subspace have constant rank defined in terms of the orders of the Galois automorphisms. Furthermore, if ordered correctly, for each integer \(k\) lying between 1 and \((n-1)/2\), the rank of any non-zero element in the sum of the first \(k\) subspaces is at most \(n-2k+1\). Slightly less sharp similar results hold for cyclic extensions of even degree.