How does 2a = 0 not imply a = 0 for a random field affect the fact that the set of n×n symmetric and skew matrices are a subspace and their dimension?

Question

For a homework assignment I'm supposed to prove that the set of n×n symmetric matrices and the set of n×n skew matrices with the elements of the matrix being elements of a random field F are a subspace and find their respective dimensions. I feel pretty good about my solution, but we received the hint that 2a = 0 does not necessarily imply a = 0, and I'm a bit concerned that this never came up. I thought maybe in proving the closure of skew matrices with elements from a field with two elements under scalar multiplication might be a problem, but since $0×1 = 0×0 = 0 = - 0$ and $1×1 = 1$ and $1×0 = 0$ and 0 and 1 are additive inverses, it seems like it still works. Is there something I'm missing?