$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$, 
$$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$

Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as:
$$
w_j(V_{G}) : =w_j({G}).
$$

>My question is that how do the  "generalized"  Stiefel-Whitney class of $O(n)$ and $SO(n)$ relate to each other? What are related conversion formulas for
$$
w_j(O(n)) = w_j(SO(n)) + ...?
$$


What I have known are that:

1. 
$$
w_3({O(3)})=w_1({O(3)})^3+w_1({O(3)})w_2({SO(3)})+w_3({SO(3)})
$$
$$
{=w_1(\mathbb{Z}_2)^3+w_1(\mathbb{Z}_2)w_2(SO(3))+w_3(SO(3)) }
$$
$$
=w_1(\mathbb{Z}_2)^3+w_1(\mathbb{Z}_2)w_2(SO(3))+w_3(SO(3)) 
$$
$$
{=w_1(\mathbb{Z}_2)^3+w_1(\mathbb{Z}_2)w_2(SO(3))+w_1(TM)w_2(O(3))}
$$
$$
{=w_1(\mathbb{Z}_2)^3+w_1(\mathbb{Z}_2)w_2(SO(3))+w_1(TM)w_2(SO(3))}
$$

2.
$$
w_2({O(3)})=w_1({O(3)})^2+w_1({O(3)})w_2({SO(3)})
$$
$$
=w_1(\mathbb{Z}_2)^2+w_1(\mathbb{Z}_2)w_2({SO(3)})
$$

3.  
When  $n=1 \mod 4$,
$$w_2(O(n)) = w_2(SO(n)) \mod 2, 
$$
When $n=3 \mod 4$,

$$
 w_2(O(n)) = w_2(SO(n)) + w_1 \cup w_1 \mod 2,
$$

The $w_1 \cup w_1$ is an obstruction to lifting $w_1$ to $\mathbb Z_4$ cohomology class.

Again, do we have 

>$$
w_j(O(n)) = w_j(SO(n)) + ...?
$$

also, do we have
>$$
w_2(O(2)) = w_2(SO(2))?
$$
$$
w_j(O(2)) = w_j(SO(2))?
$$

Edit for clarification:
p.s. See eq. 2.5 of [this journal free-access online article](https://scipost.org/SciPostPhys.4.4.021/pdf) -- I am using the same definition as theirs of “generalized” Stiefel-Whitney class of real vector bundles: $w_j(O(n))$ and $w_j(SO(n))$.