Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation
$$f(g)=\int_{\hat{G}_\mathrm{temp}}\mathrm{Trace}(\pi(\lambda(g)\check{f}))d\mu_p(\pi),$$
where $\check{f}(g)=f(g^{-1})$.

> Can one provide a reference where I can find an explicit description
> of $d\mu_p(\pi)$?

Only known case to me is for $\mathrm{SL}_2(\mathbb{R})$, where 
$$d\mu_p(\pi)=
\begin{cases}
& r\tanh(\pi r)dr, \text{ when $\pi$ is the principal series representation with parameter $ir$}\\
& nd_{\mathrm{count}}, \text{ when $\pi$ is the discrete series representation with highest weight $-n-1$}.
\end{cases}$$
I wish to have a similar description in case of $G$. Thanks in advance.