Crooked maps in Fn2

A map is called crooked if the set is an affine hyperplane for every fixed (where Fn2 is considered as a vector space over F2). We prove that the only crooked power maps are the quadratic maps xi2+j2 with gcd(n,i−j)=1. This is a consequence of the following result of independent interest: for any prime p and almost all exponents 0⩽d⩽pn−2 the set contains n linearly independent elements, where γ and a≠0 are arbitrary elements from Fpn.