Young tableaus for $SO(n)$

This is not a complete answer to this question and is adapted from my answer to Irrep decompositions for $SO(N)$ tensors for $N>3$ as it also seems to apply here. Indeed you can do decompositions and find the dimension of irreps using young tableau also for SO(n). The young tableaux describe permutation of indices and thus are relevant for all lie algebra's coming from $GL(N)$. The application to $SO(N)$ is described well in Group theory: Birdtracks, Lie's, and exceptional groups (paywalled).

The Mathematica application lieART can also do the decompositions for and find the dimensions for you for all classical and exceptional Lie algebras including $SO(N)$. The documentation in https://arxiv.org/pdf/1206.6379.pdf also explains the algorithm used to do the decomposition. The only downside is that it does not work with Young Tableaus for SO(N). This means that you will have to convert your irrep in terms of young tableau (or number of boxes per row, i.e. {4} for four index symmetric, or {1,1} for the anti-symmetric) into their Dynkin label description of the same irrep (I did not find this easy. If you know an easy way please tell).

Alternatively you can work with the description in terms of the dimensionality of the irrep but this has the downside that it is confusing when there are degeneracies in this dimension. I still found the latter easier so I converted all the young tableau to the dimension of the irrep in order to be able to feed it as input to lieART.