So let $G$ be the space of all co-dimension d subspaces. ($G$ is
naturally a manifold). I claim that if $W$ is any dimension $d$ subspace,
the set of $X$ s.t. $X \cap W = 0$ is an open, dense subset of $G$.
Open is easy. If you have $X_i$ each with non-trivial intersection with
W and $X_i \to X$ then you can pick lines $L_i$ in $W$ intersect $X_i$ and find
some limit (perhaps of a subsequence) so that $L_i \to L$. Then $L$ is in
$W \cap X$.
To show dense is also not hard. You need to show that if $X$ intersects
$W$, then you can change $X$ by a little so that it no longer does. This
really isn't hard but it is annoying to do without ever picking a
basis of anything. How about this. You can think of $X$ as the image
of a map $Y \to Z$ ($Y$ is a space of dim $n-d$, $Z$ is your big space). $X$
intersects $W$ trivially iff the map $Y \to Z/W$ is an injection. Pick
some $U$ so that $Z = U + V$. Then we can think of our map as $Y \to U+V$,
and we want the map $Y \to U$ to be injective. But $Y$ and $U$ have the same
dimension and it is easy to modify a map between such spaces by
epsilon to make it a bijection (for example add a small multiple of
some particular bijection).