Every infinitesimal rotation is a cross product

Let $A$ be 3 × 3 real matrix such that the vectors $A u$ and $u$ are orthogonal for each column vector $u \in \mathbb{R}^3$ . Prove that:

$A^{\top}=-A$ , where $A^{\top}$ denotes the transpose of the matrix $A$ ;
There exists a vector $v \in \mathbb{R}^3$ such that $A u=v \times u$ for every $u \in \mathbb{R}^3$ , where $v \times u$ denotes the vector product in $\mathbb{R}^3$ .

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Set $A=\left(a_{i j}\right), u=\left(u_1, u_2, u_3\right)^{\top}$ . If we use the orthogonality condition $(A u, u)=0$ with $u_i=\delta_{i k}$ we get $a_{k k}=0$ . If we use (1) with $u_i=\delta_{i k}+\delta_{i m}$ we get
$a_{k k}+a_{k m}+a_{m k}+a_{m m}=0$ and hence $a_{k m}=-a_{m k}$ .
Set $v_1=-a_{23}, v_2=a_{13}, v_3=-a_{12}$ . Then
$A u=\left(v_2 u_3-v_3 u_2, v_3 u_1-v_1 u_3, v_1 u_2-v_2 u_1\right)^{\top}=v \times u .$

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