In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the motivation, by Benny Cheng, the cones over them are special Lagrangians in $\mathbb{C}^{n^2+n}$. I want to know if the cones are topologically smoothable. The $n=3$ case comes from Haskins & Pacini.)

I know that for $n=3$ we are dealing with the Wu manifold, and there are many references here. For $n>3$, I tried to find references about the Stiefel-Whitney numbers of these, which by Thom, is enough for the unoriented case. However, I've only been able to locate the cohomology ring of these in Topology of Lie Groups I & II by Mimura and Toda. I don't know much algebraic topology, but I think they did not state what the Stiefel–Whitney classes are.