We can reach it by taking the rotations in three directions, e.g. a rotation around the z axis:
then we derive and evaluate at 0:
$L_{z}$ therefore represents the infinitesimal rotation.

$R_{z}(θ)=⎣⎢⎡ cos(θ)sin(θ)0 −sin(θ)cos(θ)0 001 ⎦⎥⎤ $

$L_{z}=dθdR_{z}(θ) ∣∣∣∣∣ _{0}=⎣⎢⎡ −sin(θ)cos(θ)0 −cos(θ)−sin(θ)0 001 ⎦⎥⎤ ∣∣∣∣∣ _{0}=⎣⎢⎡ 010 −100 000 ⎦⎥⎤ $

Note that the exponential map reverses this and gives a finite rotation around the Z axis back from the infinitesimal generator $L_{z}$:

$e_{θL_{z}}=R_{z}(θ)$

Repeating the same process for the other directions gives:
We have now found 3 linearly independent elements of the Lie algebra, and since $SO(3)$ has dimension 3, we are done.

$L_{x}=TODOL_{y}=TODO$

Based on the $L_{x}$,$L_{y}$ and $L_{z}$ derived at Lie algebra of $SO(3)$ we can calculate the Lie bracket as:

$TODO$

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