Most of the main results needed for this calculation can be found in Wall's paper "[Determination of the oriented cobordism ring][1]", but [this note][2] by Gwynne might be helpful to express this in more modern language. Here is the exective summary:

* All of the homotopy groups are a direct sum $\Bbb Z^r \oplus \Bbb Z/2^s$. Bordism classes of oriented manifolds are completely determined by their Chern and Stiefel-Whitney numbers.

* The mod-2 cohomology of $MSO$ is the same as the mod-2 cohomology of $BSO$, a polynomial ring on $w_2, w_3, \dots$ whose Poincare series is
$$
\prod_{i \geq 2} \tfrac{1}{1-t^i}.
$$

* Rationally, the ring is a polynomial algebra $\Bbb Q[x_4, x_8, x_{12}, \dots]$ on generators in degrees that are a power of $4$. This tells us the rank $r$ of each group. The Poincare series for the free part of $\Omega^{SO}_*$ is thus
$$
p_{free}(t) = \prod_{j \geq 1} \tfrac{1}{1-t^{4i}}.
$$

* $2$-locally, the bordism spectrum $MSO$ is a wedge of suspensions of Eilenberg--Mac Lane spectra $H\Bbb Z/2$ and $H\Bbb Z$. This allows us to write
$$
H^*(MSO) \cong \bigoplus_\text{free summands}H^*(H\Bbb Z) \oplus \bigoplus_\text{torsion summands} H^*(H\Bbb Z/2).
$$
Turning this into a Poincare series expression using the Poincare series for the cohomology of Eilenberg--Mac Lane spectra, we can solve for the Poincare series of the torsion part in $\Omega^{SO}_*$.
$$
p_{tors}(t) = \left[(1-t) \prod_{k \geq 2, k \neq 2^i-1} \left(\tfrac{1}{1-t^k}\right)\right] - \left[\frac{1}{1+t}\prod_{k \geq 1}\left(\tfrac{1}{1-t^{4k}}\right)\right]
$$

I asked Mathematica for a calculation of these groups out to degree 28. Assuming I didn't make a typo, here they are.
$$
\begin{array}{c|l}
n & \Omega^{SO}_n \\
\hline
0 & \Bbb Z\\
1 & 0\\
2 & 0\\
3 & 0\\
4 & \Bbb Z\\
5 & \Bbb Z/2\\
6 & 0\\
7 & 0\\
8 & \Bbb Z^2\\
9 & \Bbb Z/2^2\\
10 & \Bbb Z/2\\
11 & \Bbb Z/2\\
12 & \Bbb Z^3\\
13 & \Bbb Z/2^4\\
14 & \Bbb Z/2^2\\
15 & \Bbb Z/2^3\\
16 & \Bbb Z^5 \oplus \Bbb Z/2\\
17 & \Bbb Z/2^8\\
18 & \Bbb Z/2^5\\
19 & \Bbb Z/2^7\\
20 & \Bbb Z^7 \oplus \Bbb Z/2^{20}\\
21 & \Bbb Z/2^{15}\\
22 & \Bbb Z/2^{11}\\
23 & \Bbb Z/2^{15}\\
24 & \Bbb Z^{11} \oplus \Bbb Z/2^{10}\\
25 & \Bbb Z/2^{28}\\
26 & \Bbb Z/2^{22}\\
27 & \Bbb Z/2^{31}\\
28 & \Bbb Z^{15} \oplus \Bbb Z/2^{23}\\
\end{array}
$$

(The OEIS doesn't seem like anybody interested in bordism theory has invested the effort into adding this type of information.)

  [1]: https://www.maths.ed.ac.uk/~v1ranick/papers/cobord.pdf
  [2]: http://math.uchicago.edu/~may/REU2016/REUPapers/Gwynne.pdf