According to wikipedia, if $G$ is a closed subgroup of $GL(n, \mathbb{R})$ then the Lie algebra of $G$ can be thought of informally as the matrices $m$ of $M(n, \mathbb{R})$ such that $1 + εm$ is in $G$, where $ε$ is an infinitesimal positive number with $ε^2 = 0$ (of course, no such real number $ε$ exists). For example, the orthogonal group $O(n, \mathbb{R})$ consists of matrices $A$ with $A A^T = 1$, so the Lie algebra consists of the matrices $m$ with $(1 + εm)(1 + εm)^T = 1$, which is equivalent to $m + m^T = 0$ because $ε^2 = 0$.
I very much like this heuristic and am wondering if it can be used to obtain descriptions of other operations and subsets of the Lie algebra. For instance, can one derive the explicit description of the adjoint action starting with conjugation on the Lie group?