As an example, I am able to integrate $\int_{h=0}^t h dW_h$ from first principles as follows:
$$ \int_{h=0}^t h dW_h = \lim_{n\to\infty}\sum_{j=0}^{n-1}h_j(W_{h_{j+1}}-W_{h_j})=\\= \lim_{n\to\infty}\sum_{j=0}^{n-1}h_jW_{h_{j+1}}-h_jW_{h_j}+(h_{j+1}W_{h_{j+1}}-h_{j+1}W_{h_{j+1}})_{=0}=\\=\lim_{n\to\infty}\sum_{j=0}^{n-1}-W_{h_{j+1}}(h_{j+1}-h_{j})+\lim_{n\to\infty}\sum_{j=0}^{n-1}h_{j+1}W_{h_{j+1}}-h_jW_{h_j}=\\=-\int_{h=0}^tW_hdh+tW_t$$
I have tried the same approach for $\int_{h=0}^t W_h dW_h$, but cannot get the correct result:
$$ \int_{h=0}^t W_h dW_h = \lim_{n\to\infty}\sum_{j=0}^{n-1}W_{h_j}(W_{h_{j+1}}-W_{h_j})=\\=\lim_{n\to\infty}\sum_{j=0}^{n-1}W_{h_j}W_{h_{j+1}}-W_{h_j}^2 $$
I am pretty much stuck there, and don't know how to proceed to show that the integral is equal to $\frac{1}{2} W_t^2-\frac{1}{2}t$ (obviously, the result follows from Ito's lemma applied to $W_t^2$, but I would like to show it from first principles).
Any tips of hints would be greatly appreciated.
Hint
For simplification of notation, I write $W_j$ for $W_{h_j}$. Then, using $$(a-b)^2=a^2+b^2-2ab,$$ yields,
$$W_j(W_{j+1}-W_j)=\frac{1}{2}(W_{j+1}^2-W_j^2)-\frac{1}{2}(W_{j+1}-W_j)^2.$$