is the set of skew-symmetric matrices with positive Pfaffians path connected? Is the set of real $2n \times 2n$ skew-symmetric matrices having positive Pfaffians path connected?
By definition, the Pfaffian is a polynomial in the entries $a_{ij}$ ($i<j$) such that $Pf(A)^2=\det A$, and $Pf(J_n)=1$, where
$$J_n=\begin{pmatrix} 0_n & I_n \\ -I_n & 0n \end{pmatrix}.$$
 A: The answer is 'yes', because this space is the homogeneous space $\mathrm{GL}^+(2n,\mathbb{R})/\mathrm{Sp}(n,\mathbb{R})$, which is connected.
Here is more detail:  Let $J_n$ be the $2n$-by-$2n$ matrix that is $n$ diagonally placed copies of the $2$-by-$2$ matrix $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$.  Then $J_n$ is a skew-symmetric matrix with positive Pfaffian.  Any other skew-symmetric $2n$-by-$2n$ matrix $A$ with nonzero determinant can be written in the form
$$
A = aJ_na^{T}
$$ 
where $a$ is an element of $\mathrm{GL}(2n,\mathbb{R})$ that is unique up to right multiplication by an element of
$$
\mathrm{Sp}(n,\mathbb{R}) = \left\{ a\in\mathrm{GL}(2n,\mathbb{R}) \ |\ \  aJ_na^{T} = J_n\right\}.
$$
Finally, note that $\mathrm{Pf}(A) = \det(a)$, so $\mathrm{Pf}(A)>0$ if and only if $a\in\mathrm{GL}^+(2n,\mathbb{R})$.  Thus, your space of matrices is identified with the homogeneous space $\mathrm{GL}^+(2n,\mathbb{R})/\mathrm{Sp}(n,\mathbb{R})$, which is connected, since $\mathrm{GL}^+(2n,\mathbb{R})$ is connected.
Remark:  Actually, the Pfaffian is not defined as 'the' square root of $\det(A)$, as this is ambiguous.  It just happens that the Pfaffian of $A$ is a polynomial in the entries of $A$ whose square is the determinant of $A$ (which is why it is interesting).
