Derivative of the map $\mathbb{R}^9 \rightarrow \mathbb{R}^6$ $f(A)=A A^{T}-I$ Has Linearly Independent Columns When $A$ Orthogonal

Question

I am attempting to show that the derivative of the map $\mathbb{R}^9 \rightarrow \mathbb{R}^6$ $f(A)=A A^{T}-I$ has linearly independent columns when $A$ is an orthogonal matrix. I know

$$ d f_A(B)=B A^T+A B^T $$

for any 3 by 3 matrix $B$.

Taking $A\in \mathcal{O}(3)$,

$$ d f_A(B)=B A^T+A B^T=B A^{-1}+(B A^{-1})^{-1}, $$

but I can't seem to do any meaningful manipulation to this equation.

I am doing an exercise in Munkres and this is given as a hint, so that I can then apply another result to show the 3 by 3 orthogonal matrices are a manifold in $\mathbb{R}^9$.