States in Quantum Mechanics can be thought of as density operators, i.e., positive semi-definite, normalized trace class operators on a Hilbert Space $\mathcal{H}$. In the case $\mathcal{H}=\mathbb{C}^{2}$ we have that a generic state $\rho$ can be expressed as:
$$ \rho=\frac{1}{2}\left(\mathbb{I} + x^{1}\sigma_{1} + x^{2}\sigma_{2} + x^{3}\sigma_{3}\right) $$ where $\mathbb{I}$ is the identity matrix, the $\sigma_{i}$'s are the Pauli matrices and $||\,\vec{x}\,||\leq1$. This can be easily shown because $\mathcal{H}$ is finite-dimensional, and thus the set $\mathcal{B}(\mathcal{H})$ of linear operators is finite-dimensional.
I would like to know if there is a similar expression in the case in which $\mathcal{H}$ is infinite-dimensional, specifically, when $\mathcal{H}=\left(\mathcal{L}^{2}(\mathbb{R}^{n})\,;dl\right)$, where $dl$ is the Lebesgue measure.
EDIT
In order to be more specific, in the case $\mathcal{H}=\left(\mathcal{L}^{2}(\mathbb{R}^{n})\,;dl\right)$, I would like to find an expression for a generic $\rho$ suitable for the calculation of $tr(A\rho)$, where $A\in\mathcal{B}(\mathcal{H})$.
EDIT 2
In the setting I have in mind the operator $A$ is unknown. Moreover its definition depends on its action on states, in the sense that it is given as an assumption on the behaviour of $tr(A\rho)$, for example $tr(A\rho)>0$. Therefore, a generic expression for $\rho$ is needed in order to obtain informations on $A$.