Here is a sort of arbitrary example that I just picked because it is easy to construct.

Let $\{\xi_{ij}\}$ be iid $N(0,1)$ random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and let $\{a_{ij}\}$ be your favorite doubly-indexed sequence of real numbers satisfying $\sum_{i,j} |a_{ij}|^2 < \infty$.  For instance, try $a_{ij} = 2^{-i-j}$.  Then let the $ij$ entry of your random operator $T$ be $a_{ij} \xi_{ij}$.  $T$ is then almost surely Hilbert-Schmidt and in particular bounded.

More formally, let $HS$ be the Hilbert space of Hilbert-Schmidt operators on $H$, and $\|\cdot\|_{HS}$ the Hilbert-Schmidt norm $\|A\|_{HS}^2 = \sum_{i,j} |\langle A e_i, e_j \rangle|^2$.  Let $T_{ij}$ be the rank-one operator with $T e_i = e_j$ and $T e_k = 0$ for $k \ne i$. Note that $\|T_{ij}\|_{HS} = 1$.  Consider the series $T = \sum_{i,j} a_{ij} \xi_{ij} T_{ij}$.  We have
$$ \sum_{i,j} \mathbb{E} \|a_{ij} \xi_{ij} T_{ij}\|^2_{HS} = \sum_{i,j} |a_{ij}|^2 \|T_{ij}\|_{HS}^2 \mathbb{E} \xi_{ij}^2 = \sum_{i,j} |a_{ij}|^2 < \infty$$
and therefore the series converges in the vector-valued Hilbert space $L^2(\Omega; HS)$, so $T$ makes sense as an $HS$-valued random variable.  In particular, $\langle T e_i, e_j \rangle = a_{ij} \xi_{ij}$ which are independent Gaussians (and if you choose all $a_{ij}$ nonzero, they are all nondegenerate).  Now the law of $T$ is a probability measure of the kind you desire.