In the Hilbert space $L^2 (\mathbb{R})$ let the two operators $\hat x$ and $\hat p$ be defined by: $$ \hat x f(x) := x f(x),~ \hat p f(x) := -i \frac{d f(x)}{dx}. $$
Then let the commutator be defined on a maximal domain as: $$ [\hat x,\hat p] : D_{[\hat x,\hat p]} \rightarrow L^2 (\mathbb{R}), [\hat x,\hat p] := \hat x \hat p -\hat p \hat x, \\ D_{[\hat x,\hat p]} := \displaystyle{\{f(x) \in L^2 (\mathbb R) | \int_{\mathbb R} \left[(\hat x \hat p -\hat p \hat x)f(x)\right]^{*} \left[(\hat x \hat p -\hat p \hat x)f(x)\right] dx < +\infty \}}. $$
Question: Is the commutator skew-selfadjoint? If so, what is a proof of this? If not, then is it skew-essentially self-adjoint? As far as I know, the Schwartz test function space $\mathcal S (\mathbb R) $ is an essential (skew) self-adjointness domain, which is clearly a proper subset of the commutator's maximal domain, to that can't be.