I don't have the book handy, but I seem to remember that this formula is written out explicitly in S.-s Chern's Complex manifolds without potential theory (Second Edition), in the chapter "The Grassmann manifold".
Added after finding a copy of the book: Yes, as I expected, the formula the OP seeks is developed on pages 78–81 of the above source. (One may need to read a bit before this to make sure that one understands Chern's notation.)