Using spectral decomposition, we can write any symmetric matrix as
$$\Sigma = Q \Lambda Q^{\top},$$
where $Q$ is orthonormal, and
$$\Lambda = \operatorname{diag}(\lambda_1, ..., \lambda_p)$$
with $\lambda_1 \geq ... \geq \lambda_p \geq 0$.
An alternative parametrization can be made for the covariance matrix in terms of eigenvalues $\lambda_1,...,\lambda_p$ and $Q$ can be expressed using Euler angles in terms of $p(p-1)/2$ angles $\theta_{ij}$, where $i = 1,2,...,p-1$ and $j = i, ..., p-1$. [1]
Can someone elaborate on this method such that, given a function with $p$ eigenvalues and $p(p-1)/2$ angles, I can build a valid $\Sigma$.
[1]: Hoffman, Raffenetti, Ruedenberg. "Generalization of Euler Angles to N‐Dimensional Orthogonal Matrices". J. Math. Phys. 13, 528 (1972).