  We study the problem of how many different sums of squares decompositions a
general polynomial $f$ with SOS-rank $k$ admits. We show that there is a link
between the variety $\mathrm{SOS}_k(f)$ of all SOS-decompositions of $f$ and
the orthogonal group $\mathrm{O}(k)$. We exploit this connection to obtain the
dimension of $\mathrm{SOS}_k(f)$ and show that its degree is bounded from below
by the degree of $\mathrm{O}(k)$. In particular, for $k=2$ we show that
$\mathrm{SOS}_2(f)$ is isomorphic to $\mathrm{O}(2)$ and hence the degree bound
becomes an equality. Moreover, we compute the dimension of the space of
polynomials of SOS-rank $k$ and obtain the degree in the special case $k=2$.
