Dimensions of symmetric and skew-symmetric matrices

All square matrices of a given size $n$ constitute a linear space of dimension $n^2$, because to every matrix element corresponds a member of the canonical base, i.e. the set of matrices having a single $1$ and all other elements $0$.

The skew-symmetric matrices have arbitrary elements on one side with respect to the diagonal, and those elements determine the other triangle of the matrix. So they are in number of $(n^2-n)/2=n(n-1)/2$, ($-n$ to remove the diagonal).

For the symmetric matrices the reasoning is the same, but we have to add back the elements on the diagonal: $(n^2-n)/2+n=(n^2+n)/2=n(n+1)/2$.

Here is my two cents:

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\begin{eqnarray} M_{n \times n}(\mathbb{R}) & \text{has form} & \begin{pmatrix} *&*&*&*&\cdots \\ *&*&*&*& \\ *&*&*&*& \\ *&*&*&*& \\ \vdots&&&&\ddots \end{pmatrix} \hspace{.5cm} \text{with $n^2$ elements}\\ \\ \\ Skew_{n \times n}(\mathbb{R}) & \text{has form} & \begin{pmatrix} 0&*'&*'&*'&\cdots \\ *&0&*'&*'& \\ *&*&0&*'& \\ *&*&*&0& \\ \vdots&&&&\ddots \end{pmatrix} \end{eqnarray} For this bottom formation the (*)s and (*')s are just operationally inverted forms of the same number, so the array here only takes $\frac{(n^2 - n)}{2}$ elements as an argument to describe it. This appears to be an array geometry question really... I suppose if what I'm saying is true, then I conclude that because $\dim(Skew_{n \times n}(\mathbb{R}) + Sym_{n \times n}(\mathbb{R})) = \dim(M_{n \times n}(\mathbb{R}))$ and $\dim(Skew_{n \times n}(\mathbb{R}))=\frac{n^2-n}{2}$ then we have that \begin{eqnarray} \frac{n^2-n}{2}+\dim(Sym_{n \times n}(\mathbb{R})))=n^2 \end{eqnarray} or \begin{eqnarray} \dim(Sym_{n \times n}(\mathbb{R})))=\frac{n^2+n}{2}. \end{eqnarray}

This is a bit late, but I figured I would chime in since the other answers don't really give the combinatorial intuition asked for in the original question.

In order to specify a skew-symmetric matrix, we need to choose a number $A_{ij}$ for each set $\{i,j\}$ of distinct indices. How many such sets are there? Combinatorics tells us that there are $\left( \begin{array}{@{}c@{}} n \\ 2 \end{array} \right) = \frac{n(n-1)}{2}$ such sets.

Similarly, in order to specify a symmetric matrix, we need to choose a number $A_{ij}$ for each set $\{i,j\}$ of (not necessarily distinct) indices. We can encode such sets by adding a special symbol $n+1$ that indicates a repeated index. Thus, we need to choose a number $A_{ij}$ for each set $\{i,j\}$ of distinct symbols, where now a symbol is either an index ($1 , \ldots , n$), or our special symbol $n+1$ that is used for a repeated index. Combinatorics tells us that there are $\left( \begin{array}{@{}c@{}} n+1 \\ 2 \end{array} \right) = \frac{n(n+1)}{2}$ such sets.