$A$ is antisymmetric so the game itself is symmetric. Thus if player 1 can guarantee at least $v$, then player 2 can guarantee at most $-v$. Therefore, $v=0$ so you just have to find strategies $x$ and $y$ such that $x^TA=(0,0,0)$ and $Ay=(0,0,0)^T$. You get $x=y=(1/4,1/4,1/2)^T$.
Given a payoff matrix of size n, the gain of a mixed strategy $(p_1, p_2, ... p_n)$ can be expressed as a polynomial (quadratic) function of $(p_1, p_2, ... p_{n-1})$ in the polytope defined by $0\le p_i\le 1$ and $\sum_{1\le i<n}p_i\le 1$.
Finding the optimal strategies is related to find all local maximum of the function : in the general case, you need to evalute the gradient and the eigenvalues of the Hessian matrix (you can read more at https://en.wikipedia.org/wiki/Hessian_matrix#Critical_points)
If you find one maximum inside the polytope, then you have your mixed strategy (there can by only one as the function is quadratic).
If it fails (no maximum inside the polytope) then you need to find maximum on the borders (each part of the border is defined by $p_i=0$ for some $1\le i\le n$) by applying the same method with one less variable. Each border can have its own set of local maximum.
There is always the possibility to have degenerate solution (with eigenvalues = 0) that can imply that the set of optimal strategies is not a set of isolated points (but something larger, like an hyperplan $ p_1=p_2$).
In your example :
$$g(p_1,p_2)=2p_1p_2+4p_1p_3+6p_2p_3 $$ by replacing $p_3=1-p_1-p_2$
$$g(p_1,p_2)=-4p_1^2-8p_1p_2-6p_2^2+4p_1+6p_2$$
the gradient is (first derivatives) $(-8p_1-8p_2+4, -8p_1-12p_2+6)$ that is zero iff $(p_1=0, p_2=\frac{1}{2})$.
The Hessian matrix is a constant matrix (the second derivatives of a quadratic function) and here it's
$$\left[\begin{matrix}
-8 & -8 \\ -8 & -12\\
\end{matrix}\right]$$
The eigenvalues are all negative (as the determinant is positive (32) and the trace is negative (-20)), so it's a global maximum at $(0,\frac{1}{2},\frac{1}{2})$. No need to look at the borders (which can be quite complex).
So there is only one stable and optimal strategy for this game.
Best Answer
Suppose hero chooses first strategy (payoffs +8/-2) with probability $p$. We want to have the same expectation independent of which strategy the other player chooses. So
$$ 8p -4(1-p) = -2p + 20(1-p) \Rightarrow p = \frac{12}{17}.$$
gives the optimal mixed strategy.