I'm not really a math person, and apologize if the question here is too simple. I've ended up with the following type of question for a few Lie groups, but state it for SO(n).

I start with a subgroup G of SO(n) generated by rotations in $k$ fixed 2-dimensional planes $P_{1}, \ldots, P_{k}$. If I fix a permutation $\sigma \in S_{k}$, can I generate any element of $G$ that is 'sufficiently close' to the identity by multiplying rotations in the planes $P_{\sigma(1)}, \ldots, P_{\sigma(k)}$ in that order?

I know that this is nonsense for discrete groups. I don't care too much about the metric as long as it is 'reasonable' (i.e. any of the $L^{p}$ norms from being in Euclidean space, or Hilbert-Schmidt, are fine; 0-1 metric not so much). If the above factorization is possible, I'd also be interested in knowing if it ever requires going 'very far' from the origin in order to hit elements that are 'very close' to the origin.