Inserting elements in multidimensional Vector

Don't think of it as a multidimentional vector, think of it as a vector of vectors.

int n = 4;
std::vector<std::vector<int>> vec(n, std::vector<int>(n));

// looping through outer vector vec
for (int i = 0; i < n; i++) {
  // looping through inner vector vec[i]
  for (int j = 0; j < n; j++) {
    (vec[i])[j] = i*n + j;
  }
}

I included parentheses in (vec[i])[j] just for understanding.

Edit:

If you want to fill your vector via push_back, you can create a temporary vector in the inner loop, fill it, and then push_back it to your vector:

for (int i = 0; i < n; i++) {
  std::vector<int> temp_vec;

  for (int j = 0; j < n; j++) {
    temp_vec.push_back(j);
  }

  vec.push_back(temp_vec);
}

However, push_back calls result in slower code, since not only you need to reallocate your vector all the time, but also you have to create a temporary and copy it.

a vector<vector<int>> is not the best implementation for a multidimensional storage. The following implantation works for me.

template<typename T>
class array_2d {
    std::size_t data;
    std::size_t col_max;
    std::size_t row_max;
    std::vector<T> a;
public:
    array_2d(std::size_t col, std::size_t row) 
         : data(col*row), col_max(col), row_max(row), a(data)
    {}

    T& operator()(std::size_t col, std::size_t row) {
        assert(col_max > col && row_max > row)
        return a[col_max*col + row];
    }
};

use case:

array_2d<int> a(2,2);
a(0,0) = 1;
cout << a(0,0) << endl;

This solution is similar to the one described here.