Let
be the set of all symmetric
matrices and
the set of all anti-symmetric matrices (that is matrices
with
). I claim that
. Indeed, assuming that you are imposing the inner product on
by identifying it with
you can readily prove (if you're really desperate a proof can be gleaned from
here) that
from where it's immediate that
since if
then
but with equal validity
and so
. Now, to finish the argument note that every matrix
may be written as
and so
that said it's evident that
since if
is in the intersection then
. Thus,
and so
And so recalling that
you may conclude with a dimension argument