IMC 1995 Problem 7

Let $A$ be a $3\times 3$ real matrix such that the vectors $Au$ and $u$ are orthogonal for every column vector $u\in \mathbb{R}^{3}$ . Prove that: a) $A^{T}=-A$ . b) there exists a vector $v \in \mathbb{R}^{3}$ such that $Au=v\times u$ for every $u\in \mathbb{R}^{3}$ , where $v \times u$ denotes the vector product in $\mathbb{R}^{3}$ .