I am trying to prove path connectedness of $SO(n, \mathbb{R})$. I have seen several different proofs for the same. But I had a thought and wanted to know whether it would help in any way. I took two arbitrary matrices in $SO(2)$ first, one expressed by $\theta$ and another by $\phi$ of the form:
$$ \left[ \begin{array}{lr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{array} \right] $$
I then used the curve $ \sigma : [0,1] \to SO(2)$ given by $$ \sigma(t) = \left[ \begin{array}{lr} \cos(t\theta + (1-t)\phi) & \sin(t\theta + (1-t)\phi) \\ -\sin(t\theta + (1-t)\phi) & \cos(t\theta + (1-t)\phi) \end{array} \right] $$
Is this alright? Can the above argument be extended to $SO(3)$ or $SO(n)$ in general?
This is alright, essentially you use that $SO(2)$ is homeomorphoic to $S^1$. A similar method is possible for $SO(n)$, but the parametrization is a bit more involved. An inductive proof by continuously transforming one column to the standard base vector (using precisely this $n=2$ case) works best, I suggest.