I am working through Brocker & tom Dieck's Representations of Compact Lie Groups, and this is one of the early exercises. The unitary group's Lie algebra $\mathfrak{u}(n)$ is formed from skew-Hermitian matrices; the Special Unitary group's lie algebra $\mathfrak{su}(n)$ is formed from traceless antihermitian matrices. These properties seem to fall out of some basic calculations with the exponential map.
However, I am having a hard time working out a similar description for the lie algebra of $PGL(n)$ -- maybe because I do not have a good picture of the properties of its matrices to begin with. Can someone help me get a concrete idea of what $\mathfrak{pgl}(n)$ looks like?
The Lie algebra of $PGL(n,\mathbb{C})=GL(n,\mathbb{C})/Z$ is the simple Lie algebra $\mathfrak{sl}(n,\mathbb{C})$ of traceless matrices with the Lie bracket $[A,B]=AB-BA$. For references containing a proof see here.