Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?

Question

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.

Answer 1

There is a fibration $SU(n) \overset p\to SU(n)/SO(n) \overset j\to BSO(n)$, where the $j$ is the classifying map of $p$, viewed as (the projection of) a principal $SO(n)$-bundle. The Stiefel–Whitney classes for your Wu-esque manifolds are the $j^*$-images of the universal Stiefel–Whitney classes; this is in Borel and Hirzebruch's "Characteristic classes and homogeneous spaces I."

To find a nonzero Stiefel–Whitney number is to find a product of these classes of total degree $\dim SU(n)/SO(n)$. The computation in Mimura–Toda shows that the cohomology over $\mathbb F_2$ is an exterior algebra on one generator each of degrees $2$ through $n$, and that these are the images of the universal Stiefel–Whitney classes other than $w_1$. The product of these generators thus does represent the fundamental class.