Seems almost dead. Anyway, can somebody tell me if the above Bi-invariant metric has a nice expression in the local chart (V,exp) where exp is the exponential map and V is a neighborhood in so(n)(skew symmetric matrices) where exp is a diffeomorphism?

@Misha You are right, its the other way round. Am I right when I say that given a point p on So(n), the differential of the local trivialization gives a isomorphism from tangent space at p to the tangent space at (p1,p2) where p1 and p2 are points corresponding to p in S(n-1) and So(n-1) respectively? And that the tangent space at (p1,p2) is the direct sum of tangent spaces at p1 and p2? If this is true, we can use this map to define an inner product on tangent space at p(given an inner product on So(n-1)) and hence a metric on So(n). Will this metric be bi-invariant? @Robert: Thank you.

Exact map is slightly tedious to write but the quotient So(n)/So(n-1) is (n-1) sphere and we can use it to write So(n) as a fiber bundle over So(n-1) where the fiber is the (n-1) sphere. Using this, we can inductively define a metric on So(n).