How many elements of a 3x3 Rotation matrix are redundant? I read the following and got curious:

A rotation matrix is an array of nine numbers. These are subject to
  the six norm and orthogonality constraints, so only three degrees of
  freedom are left: if three of the numbers are given, the other six can
  be computed from these equations.

I know that the matrix only has three degrees of freedom, but the last statement (emphasis mine) is obviously not literally true. Given these seven elements:
$$  \left[ \begin{array}{rrrr} 
. & 0 & 0 \\
0 & . & 0 \\
0 & 0 & 1 \end{array}\right] $$
There are at least two possible solutions (1, 1 or -1, -1) which gives me valid but different rotation matrices (a rotation of 180° around the Z axis). So, if I can't choose which elements are revealed to me, I apparently need to know eight elements to be certain about the last.
Is there any way to rephrase the quote above which makes it true (are diagonal matrices an exception?). I know for instance, that if I have the following six elements: 
$$  \left[ \begin{array}{rrrr} 
a & b & c \\
d & e & f \\
. & . & . \end{array}\right] $$
I can deduce the last row by using the cross product of the first two (and choose a sign based on handedness).
Can I get away with less than these six elements?

Well, the way to best think about it is probably the following: the group of rotations $SO(n)$ is a manifold of dimension $\frac{n(n-1)}{2}$ for every positive integer $n$. This means you can construct local diffeomorphisms from $SO(n)$ to $\mathbb{R}^{\frac{n(n-1)}{2}}$, which are in particular bijective maps. For $n=3$, we have $\frac{n(n-1)}{2} = 3$, which means that (locally on $SO(3)$!) every rotation matrix is determined uniquely by 3 real numbers. This is what is meant by 3 degrees of freedom. Does this mean you can compute a unique rotation matrix given 3 real numbers? No, it doesn't, but if you have three real numbers and a diffeomorphism as above you can. However, this matrix will only be unique on the domain of the diffeomorphism.
See also https://en.wikipedia.org/wiki/Charts_on_SO(3) for details.
