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