Your conjecture is true.
The special orthogonal group $\text{SO}(n)$ is the set of all $n\times n$ matrices $R$ satisfying $R^T R=I$ and $\det(R)=1$. This is a compact lie group and its associated Lie algebra $\mathfrak{so}(n,\mathbb{R})$, which is the tangent space at the identity element, is equal to $\text{Skew}_n$, the vector space of $n\times n$ skew symmetric matrices over $\mathbb{R}$, that is real matrices satisfying $A^T=-A$. Since $\text{SO}(n)$ is a connected compact Lie group, the exponential map will be a surjective map from $\mathfrak{so}(n,\mathbb{R})$ to $\text{SO}(n)$. In other words, every element of $\text{SO}(n)$ can be written as $\exp(A)$ for a skew symmetric matrix $A$.
The vector space $\text{Skew}_n$ has the basis $\mathcal{B}=\{\mathcal{S}_{ab}:\ b>a\geq 1\}$, where the matrix $\mathcal{S}_{ab}$ has a $1$ in the $ab^{th}$ position, and a $-1$ in the $ba^{th}$ position. For example, for $n=4$, $$\mathcal{S}_{12}=\left[\begin{array}{cccc}
0 & 1 & 0 & 0\\
-1 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{array}\right].$$
Notice that this basis has cardinality $n(n-1)/2$, and that $\exp(\mathcal{S}_{ab})$ will correspond to a rotation around an axis for each $a,b$. Thus $\mathcal{SO}(n)$ is an $n(n-1)/2$ parameter family generated by rotations in $n(n-1)/2$ different planes.
