Let F be a field, and n ≥ r > 0 be integers, with r even. Denote by A n ( F ) the space of all n-by-n alternating matrices with entries in F . We consider the problem of determining the greatest possible dimension for an affine subspace of A n ( F ) in which every matrix has rank equal to r (or rank at least r). Recently Rubei [Affine subspaces of skewsymmetric matrices with constant rank. Linear Multilinear Algebra. 2023. in press. doi: 10.1080/03081087.2023.2198759.] has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that n ≥ r + 3 and | F | ≥ min ( r − 1 , r 2 + 2 ) , we also determine the affine subspaces of rank r matrices in A n ( F ) that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case n ≤ r + 2 .