linear algebra - problems of subspace of M_n(C)

let $M_n(c)$ denote the n times n matrices over the complex number field. $N$ be a subspace of

$M_n(C)$.

1 If there is no unitary lies in $N$, what is the maximum of the dimension of $N$ can be?

It's easy to see that it is not less than n(n-1), I guess it's also tight, but I don't know if I am correct.

2 If all the rank of $M$ lies in $N$ are greater than a fixed integer $k$, what is the maximum of the dimension of $N$ can be?

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