This is a generalization of the idea of Will Sawin.
The Stufe of such field should be infinite. In fact if $-1$ is a sums of
squares, i.e., $-1=a_1^2+\cdots+a_{n-1}^2$ then 
$$A:=
\begin{pmatrix}
1 & a_1&\cdots  & a_{n-1}\\
a_1 & a_1^2 &\cdots & a_1a_{n-1}\\
\vdots & \vdots&\vdots  &\vdots\\
a_{n-1} & a_{n-1}a_1&\cdots  & a_{n-1}^2\\
\end{pmatrix}
$$ would be a symmetric matrix with $A^2=0$ and is not diagonalizable.
So the base field should be a formally
real field.

A complete characterization is given in the following article (a
necessary and sufficient condition is that such field should be an
intersection of real closed fields):

D. Mornhinweg, D. B. Shapiro and K. G. Valente, The Principal Axis Theorem Over Arbitrary Fields
(The American Mathematical Monthly, Vol. 100, No. 8 (Oct., 1993), pp. 749-754.