Invertible skew-symmetric matrices have two components

Facts used. The Pfaffian is a polynomial in the entries of $K$ with $\operatorname{Pf}(K)^2=\det K$ and $\operatorname{Pf}(PKP^T)=\det(P)\operatorname{Pf}(K)$. For $J=\bigoplus_{j=1}^{n/2}\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ one has $\operatorname{Pf}(J)=1$. We also use that $GL_n^+(\mathbb{R})$ and $GL_n^-(\mathbb{R})$ are connected (standard).

Step 1: the Pfaffian separates. Since $\det K=\operatorname{Pf}(K)^2$, $K\in\mathcal S$ iff $\operatorname{Pf}(K)\ne0$. So $\mathcal S=\mathcal S_+\sqcup\mathcal S_-$ with $\mathcal S_\pm=\{\pm\operatorname{Pf}>0\}$. Each is open (Pfaffian is continuous) and they are disjoint, so there are at least two components if both are nonempty. They are nonempty: $J\in\mathcal S_+$ and $PJP^T\in\mathcal S_-$ for any $P$ with $\det P<0$.

Step 2: normal form. A real skew-symmetric matrix is orthogonally block diagonalisable: $K=Q\bigl(\bigoplus_js_j\begin{pmatrix}0&1\\-1&0\end{pmatrix}\bigr)Q^T$ with $Q\in O(n)$ and, if $K$ is invertible, $s_j>0$. Put $D=\operatorname{diag}(\sqrt{s_1},\sqrt{s_1},\sqrt{s_2},\sqrt{s_2},\dots)$. Then $\bigoplus s_j\begin{pmatrix}0&1\\-1&0\end{pmatrix}=DJD^T$, so $K=PJP^T$ with $P=QD\in GL_n(\mathbb{R})$.

Step 3: each sign set is connected. If $K\in\mathcal S_+$ then $K=PJP^T$ and $\operatorname{Pf}(K)=\det P>0$, so $P\in GL_n^+$. Hence $\mathcal S_+=\{PJP^T:P\in GL_n^+\}$ is the continuous image of the connected group $GL_n^+(\mathbb{R})$, so is connected. Similarly $\mathcal S_-=\{PJP^T:P\in GL_n^-\}$ is the image of the connected set $GL_n^-$.

Conclusion. $\mathcal S$ has exactly the two components $\mathcal S_+$ and $\mathcal S_-$.

Remark. For $n$ odd every skew matrix is singular ($\det K=\det K^T=(-1)^n\det K$), so $\mathcal S$ is empty. For $n=2$, $\mathrm{Skew}_2\cong\mathbb{R}$ via $K=\begin{pmatrix}0&a\\-a&0\end{pmatrix}$ and $\mathcal S=\mathbb{R}\setminus\{0\}$, with Pfaffian $a$: the two components are $a>0$ and $a<0$.

Lesson. The Pfaffian is the "square root of the determinant" that remembers an orientation; it detects the extra component.